From 412edaf4c7b5fb219837a60946fddf930caeba63 Mon Sep 17 00:00:00 2001
From: Patrick Stevens <patrick+github@patrickstevens.co.uk>
Date: Fri, 10 Apr 2020 19:00:57 +0100
Subject: [PATCH] Lots of refactoring towards partially-ordered ring R (#109)

---
 Everything/Safe.agda                          |    7 +-
 Fields/CauchyCompletion/Addition.agda         |    7 +-
 Fields/CauchyCompletion/Approximation.agda    |   65 +-
 Fields/CauchyCompletion/Comparison.agda       |   98 +-
 Fields/CauchyCompletion/Definition.agda       |    8 +
 Fields/CauchyCompletion/Field.agda            |   18 +-
 Fields/CauchyCompletion/Group.agda            |   78 +-
 Fields/CauchyCompletion/Multiplication.agda   |   41 +-
 .../PartiallyOrderedRing.agda                 |  160 +++
 Fields/CauchyCompletion/Ring.agda             |   45 +-
 Fields/CauchyCompletion/Setoid.agda           |   40 -
 Fields/FieldOfFractions/Order.agda            | 1134 +++++++++--------
 LogicalFormulae.agda                          |   12 +-
 Numbers/Rationals/Definition.agda             |    9 +-
 Numbers/Reals/Definition.agda                 |   12 +-
 RedBlackTree.agda                             |   42 -
 Rings/Orders/Total/Lemmas.agda                |    3 +
 Sequences.agda                                |    3 +
 Sets/Cardinality/Infinite/Examples.agda       |   11 +-
 19 files changed, 1015 insertions(+), 778 deletions(-)
 create mode 100644 Fields/CauchyCompletion/PartiallyOrderedRing.agda
 delete mode 100644 RedBlackTree.agda

diff --git a/Everything/Safe.agda b/Everything/Safe.agda
index 014ed6d..abf4a1b 100644
--- a/Everything/Safe.agda
+++ b/Everything/Safe.agda
@@ -85,6 +85,7 @@ open import Rings.Divisible.Lemmas
 open import Rings.Associates.Lemmas
 open import Rings.InitialRing
 open import Rings.Homomorphisms.Lemmas
+open import Rings.UniqueFactorisationDomains.Definition
 
 open import Setoids.Setoids
 open import Setoids.DirectSum
@@ -125,9 +126,6 @@ open import Monoids.Definition
 open import Semirings.Definition
 open import Semirings.Solver
 
-open import Fields.CauchyCompletion.Group
-open import Fields.CauchyCompletion.Ring
-
 open import Categories.Definition
 open import Categories.Functor.Definition
 open import Categories.Functor.Lemmas
@@ -149,4 +147,7 @@ open import Modules.PolynomialModule
 open import Modules.Lemmas
 open import Modules.DirectSum
 
+open import Fields.CauchyCompletion.Ring
+open import Fields.CauchyCompletion.Comparison
+
 module Everything.Safe where
diff --git a/Fields/CauchyCompletion/Addition.agda b/Fields/CauchyCompletion/Addition.agda
index f03c0ff..d7b25a5 100644
--- a/Fields/CauchyCompletion/Addition.agda
+++ b/Fields/CauchyCompletion/Addition.agda
@@ -32,9 +32,10 @@ open import Fields.CauchyCompletion.Definition order F
 open import Rings.Orders.Partial.Lemmas pRing
 open import Rings.Orders.Total.Lemmas order
 
-lemm : (m : ℕ) (a b : Sequence A) → index (apply _+_ a b) m ≡ (index a m) + (index b m)
-lemm zero a b = refl
-lemm (succ m) a b = lemm m (Sequence.tail a) (Sequence.tail b)
+private
+  lemm : (m : ℕ) (a b : Sequence A) → index (apply _+_ a b) m ≡ (index a m) + (index b m)
+  lemm zero a b = refl
+  lemm (succ m) a b = lemm m (Sequence.tail a) (Sequence.tail b)
 
 _+C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
 CauchyCompletion.elts (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) = apply _+_ a b
diff --git a/Fields/CauchyCompletion/Approximation.agda b/Fields/CauchyCompletion/Approximation.agda
index acd2801..f7d7f95 100644
--- a/Fields/CauchyCompletion/Approximation.agda
+++ b/Fields/CauchyCompletion/Approximation.agda
@@ -40,20 +40,26 @@ open import Fields.CauchyCompletion.Comparison order F charNot2
 abstract
 
   chain : {a b : A} (c : CauchyCompletion) → (a r<C c) → (c <Cr b) → a < b
-  chain {a} {b} c (betweenAC , (0<betweenAC ,, (Nac , prAC))) (betweenCB , (0<betweenCB ,, (Nb , prBC))) = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenAC a)) (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (prAC (succ Nac +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Nac)))) (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenCB (index (Sequence.tail (CauchyCompletion.elts c)) (Nac +N Nb))))))) (prBC (succ Nac +N Nb) (le Nac refl))
+  chain {a} {b} c record { e = betweenAC ; 0<e = 0<betweenAC ; N = Nac ; pr = prAC } record { e = betweenCB ; 0<e = 0<betweenCB ; N = Nb ; property = prBC } = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenAC a)) (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (prAC (succ Nac +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Nac)))) (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenCB (index (Sequence.tail (CauchyCompletion.elts c)) (Nac +N Nb))))))) (prBC (succ Nac +N Nb) (le Nac refl))
 
-  approxLemma : (a : CauchyCompletion) (e e/2 : A) → (0G < e) → (e/2 + e/2 ∼ e) → (m N : ℕ) → abs ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts a) N)) < e/2 → (e/2 + index (CauchyCompletion.elts a) m) < (index (CauchyCompletion.elts a) N + e)
-  approxLemma a e e/2 0<e prE/2 m N ans with totality 0R ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts a) N))
-  approxLemma a e e/2 0<e prE/2 m N ans | inl (inl x) with <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (invLeft))) identRight) groupIsAbelian (orderRespectsAddition ans (index (CauchyCompletion.elts a) N))
-  ... | bl = <WellDefined groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) prE/2)) (orderRespectsAddition bl e/2)
-  approxLemma a e e/2 0<e prE/2 m N ans | inl (inr x) with <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (invLeft))) identRight) (identLeft) (orderRespectsAddition x (index (CauchyCompletion.elts a) N))
-  ... | bl = <WellDefined groupIsAbelian (Equivalence.reflexive eq) (ringAddInequalities bl (halfLess e/2 e 0<e prE/2))
-  approxLemma a e e/2 0<e prE/2 m N ans | inr x with transferToRight additiveGroup (Equivalence.symmetric eq x)
-  ... | bl = <WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bl)) groupIsAbelian (orderRespectsAddition (halfLess e/2 e 0<e prE/2) (index (CauchyCompletion.elts a) N))
+  private
+    approxLemma : (a : CauchyCompletion) (e e/2 : A) → (0G < e) → (e/2 + e/2 ∼ e) → (m N : ℕ) → abs ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts a) N)) < e/2 → (e/2 + index (CauchyCompletion.elts a) m) < (index (CauchyCompletion.elts a) N + e)
+    approxLemma a e e/2 0<e prE/2 m N ans with totality 0R ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts a) N))
+    approxLemma a e e/2 0<e prE/2 m N ans | inl (inl x) with <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (invLeft))) identRight) groupIsAbelian (orderRespectsAddition ans (index (CauchyCompletion.elts a) N))
+    ... | bl = <WellDefined groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) prE/2)) (orderRespectsAddition bl e/2)
+    approxLemma a e e/2 0<e prE/2 m N ans | inl (inr x) with <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (invLeft))) identRight) (identLeft) (orderRespectsAddition x (index (CauchyCompletion.elts a) N))
+    ... | bl = <WellDefined groupIsAbelian (Equivalence.reflexive eq) (ringAddInequalities bl (halfLess e/2 e 0<e prE/2))
+    approxLemma a e e/2 0<e prE/2 m N ans | inr x with transferToRight additiveGroup (Equivalence.symmetric eq x)
+    ... | bl = <WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bl)) groupIsAbelian (orderRespectsAddition (halfLess e/2 e 0<e prE/2) (index (CauchyCompletion.elts a) N))
 
-  approximateAboveCrude : (a : CauchyCompletion) → Sg A (λ b → (a <Cr b))
-  approximateAboveCrude a with CauchyCompletion.converges a 1R (0<1 (charNot2ImpliesNontrivial R charNot2))
-  ... | N , conv = ((((index (CauchyCompletion.elts a) (succ N)) + 1R) + 1R) + 1R) , (1R , (0<1 (charNot2ImpliesNontrivial R charNot2) ,, (N , ans)))
+approximateAboveCrude : (a : CauchyCompletion) → A
+approximateAboveCrude a with CauchyCompletion.converges a 1R (0<1 (charNot2ImpliesNontrivial R charNot2))
+... | N , conv = (((index (CauchyCompletion.elts a) (succ N)) + 1R) + 1R) + 1R
+
+abstract
+  approximateAboveCrudeIsAbove : (a : CauchyCompletion) → a <Cr (approximateAboveCrude a)
+  approximateAboveCrudeIsAbove a with CauchyCompletion.converges a 1R (0<1 (charNot2ImpliesNontrivial R charNot2))
+  ... | N , conv = record { e = 1R ; 0<e = 0<1 (charNot2ImpliesNontrivial R charNot2) ; N = N ; property = ans }
     where
       ans : (m : ℕ) → (N <N m) → (1R + index (CauchyCompletion.elts a) m) < (((index (CauchyCompletion.elts a) (succ N) + 1R) + 1R) + 1R)
       ans m N<m with conv {m} {succ N} N<m (le 0 refl)
@@ -64,14 +70,15 @@ abstract
       ... | am<aN = <WellDefined groupIsAbelian (Equivalence.reflexive eq) (orderRespectsAddition (SetoidPartialOrder.<Transitive pOrder am<aN (<WellDefined (Equivalence.reflexive eq) (+Associative) (<WellDefined identLeft groupIsAbelian (orderRespectsAddition (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (0<1 (charNot2ImpliesNontrivial R charNot2)) (0<1 (charNot2ImpliesNontrivial R charNot2)))) (index (CauchyCompletion.elts a) (succ N)))))) 1R)
       ans m N<m | bl | inr 0=am-aN = <WellDefined (Equivalence.transitive eq (+WellDefined identLeft (Equivalence.reflexive eq)) identLeft) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq groupIsAbelian (+WellDefined (transferToRight additiveGroup (Equivalence.symmetric eq 0=am-aN)) (Equivalence.reflexive eq))) (Equivalence.reflexive eq)) +Associative)) (orderRespectsAddition (ringAddInequalities (0<1 (charNot2ImpliesNontrivial R charNot2)) (0<1 (charNot2ImpliesNontrivial R charNot2))) (1R + (index (CauchyCompletion.elts a) m)))
 
-  rationalApproximatelyAbove : (a : CauchyCompletion) → (e : A) → (0G < e) → A
-  rationalApproximatelyAbove a e 0<e with halve charNot2 e
-  ... | e/2 , prE/2 with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e)
-  ... | 0<e/2 with halve charNot2 e/2
-  ... | e/4 , prE/4 with halvePositive e/4 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/4) 0<e/2)
-  ... | 0<e/4 with CauchyCompletion.converges a e/4 0<e/4
-  ... | N , cauchyBeyondN = index (CauchyCompletion.elts a) (succ N) + e/2
+rationalApproximatelyAbove : (a : CauchyCompletion) → (e : A) → (0G < e) → A
+rationalApproximatelyAbove a e 0<e with halve charNot2 e
+... | e/2 , prE/2 with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e)
+... | 0<e/2 with halve charNot2 e/2
+... | e/4 , prE/4 with halvePositive e/4 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/4) 0<e/2)
+... | 0<e/4 with CauchyCompletion.converges a e/4 0<e/4
+... | N , cauchyBeyondN = index (CauchyCompletion.elts a) (succ N) + e/2
 
+abstract
   rationalApproximatelyAboveIsAbove : (a : CauchyCompletion) (e : A) → (0<e : 0G < e) → a <Cr (rationalApproximatelyAbove a e 0<e)
   rationalApproximatelyAboveIsAbove a e 0<e with halve charNot2 e
   ... | e/2 , prE/2 with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e)
@@ -81,7 +88,7 @@ abstract
   ... | N , cauchyBeyondN with halve charNot2 e/4
   ... | e/8 , prE/8 with halvePositive e/8 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/8) 0<e/4)
   ... | 0<e/8 with CauchyCompletion.converges a e/8 0<e/8
-  ... | N2 , cauchyBeyondN2 = e/8 , (0<e/8 ,, ((N +N N2) , ans2))
+  ... | N2 , cauchyBeyondN2 = record { e = e/8 ; 0<e = 0<e/8 ; N = N +N N2 ; property = ans2 }
     where
       ans2 : (m : ℕ) → N +N N2 <N m → (e/8 + index (CauchyCompletion.elts a) m) < (index (CauchyCompletion.elts a) (succ N) + e/2)
       ans2 m <m with cauchyBeyondN {m} {succ N} (inequalityShrinkLeft <m) (le 0 refl)
@@ -105,7 +112,7 @@ abstract
   ... | 0<e/8 with CauchyCompletion.converges a e/8 0<e/8
   ... | N8 , cauchyBeyondN8 with halve charNot2 e/8
   ... | e/16 , prE/16 with halvePositive e/16 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/16) 0<e/8)
-  ... | 0<e/16 = ((e/2 + e/4) + e/8) , ((e/8 , (0<e/8 ,, ((N +N N8) , ans))) ,, (e/16 , (0<e/16 ,, (0 , t'))))
+  ... | 0<e/16 = record { i = (e/2 + e/4) + e/8 ; a<i = record { e = e/8 ; 0<e = 0<e/8 ; N = N +N N8 ; property = ans } ; i<b = record { e = e/16 ; 0<e = 0<e/16 ; N = 0 ; pr = t' } }
     where
       t' : (m : ℕ) → (0 <N m) → (((e/2 + e/4) + e/8) + e/16) < index (constSequence e) m
       t' m 0<m rewrite indexAndConst e m = <WellDefined (Equivalence.reflexive eq) prE/2 (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) prE/4) (<WellDefined +Associative (Equivalence.symmetric eq +Associative) (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (<WellDefined (Equivalence.reflexive eq) prE/8 (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (halfLess e/16 e/8 0<e/8 prE/16) e/8))) (e/2 + e/4)))))
@@ -130,7 +137,7 @@ abstract
 
   approximateBelow : (a : CauchyCompletion) → (ε : A) → (0G < ε) → Sg A (λ b → (b r<C a) && (a +C (-C injection b)) <C (injection ε))
   approximateBelow a e 0<e with approximateAbove (-C a) e 0<e
-  ... | x , ((deltaXAnd-A , (0<deltaXA ,, (NdeltaXA , prDeltaXA))) ,, (rationalNear , ((bound , (0<bound ,, (N , prBound))) ,, (bound2 , (0<bound2 ,, (N2 , prBound2)))))) = inverse x , ((deltaXAnd-A , (0<deltaXA ,, (NdeltaXA , λ m N<m → <WellDefined (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (cancel {m} (CauchyCompletion.elts a))) identRight)))))) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invRight (Equivalence.reflexive eq)) identLeft)) (orderRespectsAddition (prDeltaXA m N<m) (inverse x + (index (CauchyCompletion.elts a) m)))))) ,, (rationalNear , ((bound , (0<bound ,, (N , pr1))) ,, (bound2 , (0<bound2 ,, (N2 , prBound2))))))
+  ... | x , (record { e = deltaXAnd-A ; 0<e = 0<deltaXA ; N = NdeltaXA ; property = prDeltaXA } ,, record { i = rationalNear ; a<i = record { e = bound ; 0<e = 0<bound ; N = N ; property = prBound } ; i<b = record { e = bound2 ; 0<e = 0<bound2 ; N = N2 ; pr = prBound2 } }) = inverse x , (record { e = deltaXAnd-A ; 0<e = 0<deltaXA ; N = NdeltaXA ; pr = λ m N<m → <WellDefined (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (cancel {m} (CauchyCompletion.elts a))) identRight)))))) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invRight (Equivalence.reflexive eq)) identLeft)) (orderRespectsAddition (prDeltaXA m N<m) (inverse x + (index (CauchyCompletion.elts a) m))) } ,, record { i = rationalNear ; a<i = record { e = bound ; 0<e = 0<bound ; N = N ; property = pr1 } ; i<b = record { e = bound2 ; 0<e = 0<bound2 ; N = N2 ; pr = prBound2 } })
     where
       cancel : {m : ℕ} (a : Sequence A) → (index (map inverse a) m) + (index a m) ∼ 0G
       cancel {m} a rewrite equalityCommutative (mapAndIndex a inverse m) = invLeft
@@ -143,7 +150,7 @@ abstract
   ... | below , (below<a ,, a-below<e) with approximateAbove a 1R (0<1 (charNot2ImpliesNontrivial R charNot2))
   ... | above , (a<above ,, above-a<e) with totality 0R below
   boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inl 0<below) with totality 0R above
-  boundModulus a | below , ((belowBound , (0<belowBound ,, (Nbelow , prBelow))) ,, a-below<e) | above , ((bound , (0<bound ,, (N , ans))) ,, above-a<e) | inl (inl 0<below) | inl (inl 0<above) = above , ((N +N Nbelow) , λ m N<m → SetoidPartialOrder.<Transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
+  boundModulus a | below , (record { e = belowBound ; 0<e = 0<belowBound ; N = Nbelow ; pr = prBelow } ,, a-below<e) | above , (record { e = bound ; 0<e = 0<bound ; N = N ; property = ans } ,, above-a<e) | inl (inl 0<below) | inl (inl 0<above) = above , ((N +N Nbelow) , λ m N<m → SetoidPartialOrder.<Transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
     where
       res : (m : ℕ) → ((N +N Nbelow) <N m) → (abs (index (CauchyCompletion.elts a) m)) < (bound + index (CauchyCompletion.elts a) m)
       res m N<m with totality 0R (index (CauchyCompletion.elts a) m)
@@ -154,42 +161,42 @@ abstract
   boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inl 0<below) | inr 0=above = exFalso (irreflexive (SetoidPartialOrder.<Transitive pOrder 0<below (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=above) (chain a below<a a<above))))
   boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inr below<0) with totality 0R above
   boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) with totality (inverse below) above
-  boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inl (inl -bel<ab) = above , ((N +N Nabove) , ans)
+  boundModulus a | below , (record { e = boundBelow ; 0<e = 0<boundBelow ; N = N ; pr = prBoundBelow } ,, a-below<e) | above , (record { e = boundAbove ; 0<e = 0<boundAbove ; N = Nabove ; property = prBoundAbove } ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inl (inl -bel<ab) = above , ((N +N Nabove) , ans)
     where
       ans : (m : ℕ) → (N +N Nabove <N m) → abs (index (CauchyCompletion.elts a) m) < above
       ans m N<m with totality 0G (index (CauchyCompletion.elts a) m)
       ans m N<m | inl (inl 0<am) = SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))
       ans m N<m | inl (inr am<0) = SetoidPartialOrder.<Transitive pOrder (ringSwapNegatives' (prBoundBelow m (inequalityShrinkLeft N<m))) (SetoidPartialOrder.<Transitive pOrder (ringSwapNegatives' (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below))) -bel<ab)
       ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) 0<above
-  boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inl (inr ab<-bel) = inverse below , ((N +N Nabove) , ans)
+  boundModulus a | below , (record { e = boundBelow ; 0<e = 0<boundBelow ; N = N ; pr = prBoundBelow } ,, a-below<e) | above , (record { e = boundAbove ; 0<e = 0<boundAbove ; N = Nabove ; property = prBoundAbove } ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inl (inr ab<-bel) = inverse below , ((N +N Nabove) , ans)
     where
       ans : (m : ℕ) → (N +N Nabove <N m) → abs (index (CauchyCompletion.elts a) m) < (inverse below)
       ans m N<m with totality 0G (index (CauchyCompletion.elts a) m)
       ans m N<m | inl (inl 0<am) = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) ab<-bel
       ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
       ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) (lemm2 below below<0)
-  boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inr -bel=ab = above , ((N +N Nabove) , ans)
+  boundModulus a | below , (record { e = boundBelow ; 0<e = 0<boundBelow ; N = N ; pr = prBoundBelow } ,, a-below<e) | above , (record { e = boundAbove ; 0<e = 0<boundAbove ; N = Nabove ; property = prBoundAbove } ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inr -bel=ab = above , ((N +N Nabove) , ans)
     where
       ans : (m : ℕ) → (N +N Nabove <N m) → abs (index (CauchyCompletion.elts a) m) < above
       ans m N<m with totality 0G (index (CauchyCompletion.elts a) m)
       ans m N<m | inl (inl 0<am) = SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))
       ans m N<m | inl (inr am<0) = <WellDefined (Equivalence.reflexive eq) (-bel=ab) (ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m))))
       ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) 0<above
-  boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inr above<0) = inverse below , ((N +N Nabove) , ans)
+  boundModulus a | below , (record { e = boundBelow ; 0<e = 0<boundBelow ; N = N ; pr = prBoundBelow } ,, a-below<e) | above , (record { e = boundAbove ; 0<e = 0<boundAbove ; N = Nabove ; property = prBoundAbove } ,, above-a<e) | inl (inr below<0) | inl (inr above<0) = inverse below , ((N +N Nabove) , ans)
     where
       ans : (m : ℕ) → ((N +N Nabove) <N m) → abs (index (CauchyCompletion.elts a) m) < inverse below
       ans m N<m with totality 0R (index (CauchyCompletion.elts a) m)
       ans m N<m | inl (inl 0<am) = exFalso (irreflexive {0G} (SetoidPartialOrder.<Transitive pOrder 0<am (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) above<0)))
       ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
       ans m N<m | inr 0=am = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=am) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) above<0)))
-  boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inr 0=above = inverse below , ((N +N Nabove) , ans)
+  boundModulus a | below , (record { e = boundBelow ; 0<e = 0<boundBelow ; N = N ; pr = prBoundBelow } ,, a-below<e) | above , (record { e = boundAbove ; 0<e = 0<boundAbove ; N = Nabove ; property = prBoundAbove } ,, above-a<e) | inl (inr below<0) | inr 0=above = inverse below , ((N +N Nabove) , ans)
     where
       ans : (m : ℕ) → ((N +N Nabove) <N m) → abs (index (CauchyCompletion.elts a) m) < inverse below
       ans m N<m with totality 0R (index (CauchyCompletion.elts a) m)
       ans m N<m | inl (inl 0<am) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=above) (SetoidPartialOrder.<Transitive pOrder 0<am (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))))))
       ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
       ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) (lemm2 _ below<0)
-  boundModulus a | below , ((boundBelow , ((boundBelowDiff ,, (Nb , ansBelow)))) ,, a-below<e) | above , ((bound , (0<bound ,, (N , ans))) ,, above-a<e) | inr 0=below = above , ((N +N Nb) , λ m N<m → SetoidPartialOrder.<Transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
+  boundModulus a | below , (record { e = boundBelow ; 0<e = boundBelowDiff ; N = Nb ; pr = ansBelow } ,, a-below<e) | above , (record { e = bound ; 0<e = 0<bound ; N = N ; property = ans } ,, above-a<e) | inr 0=below = above , ((N +N Nb) , λ m N<m → SetoidPartialOrder.<Transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
     where
       res : (m : ℕ) → (N +N Nb) <N m → (abs (index (CauchyCompletion.elts a) m)) < (bound + index (CauchyCompletion.elts a) m)
       res m N<m with totality 0R (index (CauchyCompletion.elts a) m)
diff --git a/Fields/CauchyCompletion/Comparison.agda b/Fields/CauchyCompletion/Comparison.agda
index aff362b..1d04c4c 100644
--- a/Fields/CauchyCompletion/Comparison.agda
+++ b/Fields/CauchyCompletion/Comparison.agda
@@ -15,6 +15,7 @@ open import Functions
 open import LogicalFormulae
 open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Order
+open import Numbers.Naturals.Order.Lemmas
 open import Semirings.Definition
 
 module Fields.CauchyCompletion.Comparison {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
@@ -35,18 +36,16 @@ open import Fields.Lemmas F
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Setoid order F charNot2
 
-shrinkInequality : {a b c : ℕ} → a +N b <N c → a <N c
-shrinkInequality {a} {b} {c} (le x pr) = le (x +N b) (transitivity (applyEquality succ (transitivity (equalityCommutative (Semiring.+Associative ℕSemiring x _ _)) (applyEquality (x +N_) (Semiring.commutative ℕSemiring b a)))) pr)
-
-shrinkInequality' : {a b c : ℕ} → a +N b <N c → b <N c
-shrinkInequality' {a} {b} {c} (le x pr) = le (x +N a) (transitivity (applyEquality succ (equalityCommutative (Semiring.+Associative ℕSemiring x _ _))) pr)
-
 -- "<C rational", where the r indicates where the rational number goes
-_<Cr_ : CauchyCompletion → A → Set (m ⊔ o)
-a <Cr b = Sg A (λ ε → (0G < ε) && Sg ℕ (λ N → ((m : ℕ) → (N<m : N <N m) → (ε + index (CauchyCompletion.elts a) m) < b)))
+record _<Cr_ (a : CauchyCompletion) (b : A) : Set (m ⊔ o) where
+  field
+    e : A
+    0<e : 0G < e
+    N : ℕ
+    property : ((m : ℕ) → (N<m : N <N m) → (e + index (CauchyCompletion.elts a) m) < b)
 
 <CrWellDefinedRight : (a : CauchyCompletion) (b c : A) → b ∼ c → a <Cr b → a <Cr c
-<CrWellDefinedRight a b c b=c (ε , (0<e ,, (N , pr))) = ε , (0<e ,, (N , λ m N<m → <WellDefined (Equivalence.reflexive eq) b=c (pr m N<m)))
+<CrWellDefinedRight a b c b=c record { e = ε ; 0<e = 0<e ; N = N ; property = pr } = record { e = ε ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (Equivalence.reflexive eq) b=c (pr m N<m) }
 
 <CrWellDefinedLemma : (a b e/2 e : A) (0<e : 0G < e) (pr : e/2 + e/2 ∼ e) → abs (a + inverse b) < e/2 → (e/2 + b) < (e + a)
 <CrWellDefinedLemma a b e/2 e 0<e pr a-b<e with totality 0G (a + inverse b)
@@ -55,23 +54,27 @@ a <Cr b = Sg A (λ ε → (0G < ε) && Sg ℕ (λ N → ((m : ℕ) → (N<m : N
 <CrWellDefinedLemma a b e/2 e 0<e pr a-b<e | inr 0=a-b = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (transferToRight (Ring.additiveGroup R) (Equivalence.symmetric eq 0=a-b)))) (orderRespectsAddition (halfLess e/2 e 0<e pr) b)
 
 <CrWellDefinedLeft : (a b : CauchyCompletion) (c : A) → Setoid._∼_ cauchyCompletionSetoid a b → a <Cr c → b <Cr c
-<CrWellDefinedLeft a b c a=b (ε , (0<e ,, (N , pr))) with halve charNot2 ε
+<CrWellDefinedLeft a b c a=b record { e = ε ; 0<e = 0<e ; N = N ; property = pr } with halve charNot2 ε
 ... | e/2 , prE/2 with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e)
 ... | 0<e/2 with a=b e/2 0<e/2
-... | N2 , prN2 = e/2 , (0<e/2 ,, ((N +N N2) , ans))
+... | N2 , prN2 = record { e = e/2 ; 0<e = 0<e/2 ; N = N +N N2 ; property = ans }
   where
     ans : (m : ℕ) → (N<m : (N +N N2) <N m) → (e/2 + index (CauchyCompletion.elts b) m) < c
-    ans m N<m with prN2 {m} (shrinkInequality' N<m)
-    ... | bl rewrite indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts b) inverse m) = SetoidPartialOrder.<Transitive pOrder (<CrWellDefinedLemma (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) e/2 ε 0<e prE/2 bl) (pr m (shrinkInequality N<m))
+    ans m N<m with prN2 {m} (inequalityShrinkRight N<m)
+    ... | bl rewrite indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts b) inverse m) = SetoidPartialOrder.<Transitive pOrder (<CrWellDefinedLemma (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) e/2 ε 0<e prE/2 bl) (pr m (inequalityShrinkLeft N<m))
 
 <CrWellDefined : (a b : CauchyCompletion) (c d : A) → Setoid._∼_ cauchyCompletionSetoid a b → c ∼ d → a <Cr c → b <Cr d
 <CrWellDefined a b c d a=b c=d a<c = <CrWellDefinedLeft a b d a=b (<CrWellDefinedRight a c d c=d a<c)
 
-_r<C_ : A → CauchyCompletion → Set (m ⊔ o)
-a r<C b = Sg A (λ ε → (0G < ε) && Sg ℕ (λ N → ((m : ℕ) → (N<m : N <N m) → (a + ε) < index (CauchyCompletion.elts b) m)))
+record _r<C_ (a : A) (b : CauchyCompletion) : Set (m ⊔ o) where
+  field
+    e : A
+    0<e : 0G < e
+    N : ℕ
+    pr : (m : ℕ) → (N<m : N <N m) → (a + e) < index (CauchyCompletion.elts b) m
 
 r<CWellDefinedLeft : (a b : A) (c : CauchyCompletion) → (a ∼ b) → (a r<C c) → b r<C c
-r<CWellDefinedLeft a b c a=b (e , (0<e ,, (N , pr))) = e , (0<e ,, (N , λ m N<m → <WellDefined (+WellDefined a=b (Equivalence.reflexive eq)) (Equivalence.reflexive eq) (pr m N<m)))
+r<CWellDefinedLeft a b c a=b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } = record { e = e ; 0<e = 0<e ; N = N ; pr =  λ m N<m → <WellDefined (+WellDefined a=b (Equivalence.reflexive eq)) (Equivalence.reflexive eq) (pr m N<m) }
 
 r<CWellDefinedLemma : (a b c e e/2 : A) (_ : e/2 + e/2 ∼ e) (0<e : 0G < e) (_ : abs (a + inverse b) < e/2) (_ : (c + e) < a) → (c + e/2) < b
 r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a with totality 0G (a + inverse b)
@@ -80,49 +83,72 @@ r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inl (inr a-b<0) = SetoidPar
 r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inr 0=a-b = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (orderRespectsAddition (halfLess e/2 e 0<e prE/2) c)) (<WellDefined (groupIsAbelian {c} {e}) (transferToRight additiveGroup (Equivalence.symmetric eq 0=a-b)) c+e<a)
 
 r<CWellDefinedRight : (a b : CauchyCompletion) (c : A) → Setoid._∼_ cauchyCompletionSetoid a b → c r<C a → c r<C b
-r<CWellDefinedRight a b c a=b (e , (0<e ,, (N , pr))) with halve charNot2 e
+r<CWellDefinedRight a b c a=b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } with halve charNot2 e
 ... | e/2 , prE/2 with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e)
 ... | 0<e/2 with a=b e/2 0<e/2
-... | N2 , prN2 = e/2 , (0<e/2 ,, ((N +N N2) , ans))
+... | N2 , prN2 = record { e = e/2 ; 0<e = 0<e/2 ; N = N +N N2 ; pr = ans }
   where
     ans : (m : ℕ) → (N<m : (N +N N2) <N m) → (c + e/2) < index (CauchyCompletion.elts b) m
-    ans m N<m with prN2 {m} (shrinkInequality' N<m)
-    ... | bl rewrite indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts b) inverse m) = r<CWellDefinedLemma (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) c e e/2 prE/2 0<e bl (pr m (shrinkInequality N<m))
+    ans m N<m with prN2 {m} (inequalityShrinkRight N<m)
+    ... | bl rewrite indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts b) inverse m) = r<CWellDefinedLemma (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) c e e/2 prE/2 0<e bl (pr m (inequalityShrinkLeft N<m))
 
-_<C_ : CauchyCompletion → CauchyCompletion → Set (m ⊔ o)
-a <C b = Sg A (λ i → (a <Cr i) && (i r<C b))
+record _<C_ (a : CauchyCompletion) (b : CauchyCompletion) : Set (m ⊔ o) where
+  field
+    i : A
+    a<i : a <Cr i
+    i<b : i r<C b
 
 <CWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → a <C c → b <C d
-<CWellDefined {a} {b} {c} {d} a=b c=d (bound , (a<bound ,, bound<b)) = bound , (<CrWellDefinedLeft a b bound a=b a<bound ,, r<CWellDefinedRight c d bound c=d bound<b)
+<CWellDefined {a} {b} {c} {d} a=b c=d record { i = bound ; a<i = a<bound ; i<b = bound<b } = record { i = bound ; a<i = <CrWellDefinedLeft a b bound a=b a<bound ; i<b = r<CWellDefinedRight c d bound c=d bound<b }
 
-<CrPreserves : {a b : A} → a < b → injection a <Cr b
-<CrPreserves {a} {b} a<b with dense charNot2 a<b
-... | between , (a<between ,, between<b) = (between + inverse a) , (<WellDefined invRight (Equivalence.reflexive eq) (orderRespectsAddition a<between (inverse a)) ,, (0 , λ m _ → <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (ans a m)) +Associative)) (Equivalence.reflexive eq) between<b))
+<CInheritedL : {a b : A} → a < b → injection a <Cr b
+<CInheritedL {a} {b} a<b with dense charNot2 a<b
+... | between , (a<between ,, between<b) = record { e = between + inverse a ; 0<e = <WellDefined invRight (Equivalence.reflexive eq) (orderRespectsAddition a<between (inverse a)) ; N = 0 ; property = λ m _ → <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (ans a m)) +Associative)) (Equivalence.reflexive eq) between<b }
   where
     ans : (x : A) (m : ℕ) → 0G ∼ inverse x + index (constSequence x) m
     ans x m rewrite indexAndConst x m = Equivalence.symmetric eq invLeft
 
-r<CPreserves : {a b : A} → a < b → a r<C injection b
-r<CPreserves {a} {b} a<b with dense charNot2 a<b
-... | between , (a<between ,, between<b) = (between + inverse a) , (<WellDefined invRight (Equivalence.reflexive eq) (orderRespectsAddition a<between (inverse a)) ,, (0 , λ m _ → <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq invLeft))) +Associative) groupIsAbelian) (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (indexAndConst b m)) between<b))
+<CInheritedR : {a b : A} → a < b → a r<C injection b
+<CInheritedR {a} {b} a<b with dense charNot2 a<b
+... | between , (a<between ,, between<b) = record { e = between + inverse a ; 0<e = <WellDefined invRight (Equivalence.reflexive eq) (orderRespectsAddition a<between (inverse a)) ; N = 0 ; pr = λ m _ → <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq invLeft))) +Associative) groupIsAbelian) (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (indexAndConst b m)) between<b }
 
 <CInherited : {a b : A} → a < b → (injection a) <C (injection b)
 <CInherited {a} {b} a<b with dense charNot2 a<b
-... | between , (a<between ,, between<b) = between , (<CrPreserves a<between ,, r<CPreserves between<b)
+... | between , (a<between ,, between<b) = record { i = between ; a<i = <CInheritedL a<between ; i<b = <CInheritedR between<b }
 
-<CInherited' : {a b : A} → (injection a) <C (injection b) → a < b
-<CInherited' {a} {b} (bound , ((Na , (0<Na ,, (Na2 , prA))) ,, ((Nb , (0<Nb ,, (Nb2 , prB)))))) with prA (succ Na2) (le 0 refl)
-... | bl with prB (succ Nb2) (le 0 refl)
-... | bl2 rewrite indexAndConst a Na2 | indexAndConst b Nb2 = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<Na a)) bl) (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<Nb bound)) bl2)
+<CCollapsesL : (a b : A) → (injection a) <Cr b → a < b
+<CCollapsesL a b record { e = e ; 0<e = 0<i ; N = N ; property = pr } with pr (succ N) (le 0 refl)
+... | bl rewrite indexAndConst a N = <Transitive (<WellDefined identLeft reflexive (orderRespectsAddition 0<i a)) bl
+
+<CCollapsesR : (a b : A) → a r<C injection b → a < b
+<CCollapsesR a b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } with pr (succ N) (le 0 refl)
+... | bl rewrite indexAndConst b N = <Transitive (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<e a)) bl
+
+<CCollapses : {a b : A} → (injection a) <C (injection b) → a < b
+<CCollapses {a} {b} record { i = inter ; a<i = pr1 ; i<b = pr2 } = <Transitive (<CCollapsesL a inter pr1) (<CCollapsesR inter b pr2)
+
+<CRelaxL : {a : A} {b : CauchyCompletion} → a r<C b → injection a <C b
+<CRelaxL {a} {b} record { e = e ; 0<e = 0<e ; N = N ; pr = pr } with halve charNot2 e
+... | e/2 , e/2pr = record { i = a + e/2 ; a<i = <CInheritedL (<WellDefined identLeft groupIsAbelian (orderRespectsAddition (halvePositive' e/2pr 0<e) a)) ; i<b = record { e = e/2 ; 0<e = halvePositive' e/2pr 0<e ; N = N ; pr = λ m N<m → <WellDefined (transitive (+WellDefined reflexive (symmetric e/2pr)) +Associative) reflexive (pr m N<m) } }
+
+<CRelaxL' : {a : A} {b : CauchyCompletion} → injection a <C b → a r<C b
+<CRelaxL' {a} {b} record { i = i ; a<i = a<i ; i<b = record { e = e ; 0<e = 0<e ; N = N ; pr = pr } } = record { e = e ; 0<e = 0<e ; N = N ; pr = λ m N<m → <Transitive (orderRespectsAddition (<CCollapsesL _ _ a<i) e) (pr m N<m) }
+
+<CRelaxR : {a : CauchyCompletion} {b : A} → a <Cr b → a <C injection b
+<CRelaxR {a} {b} record { e = ε ; 0<e = 0<ε ; N = N ; property = property } with halve charNot2 ε
+... | e/2 , e/2+e/2=e = record { i = b + inverse e/2 ; a<i = record { e = e/2 ; 0<e = halvePositive' e/2+e/2=e 0<ε ; N = N ; property = λ m N<m → <WellDefined (transitive groupIsAbelian (transitive +Associative (+WellDefined (transitive (+WellDefined reflexive (symmetric e/2+e/2=e)) (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft))) reflexive))) reflexive (orderRespectsAddition (property m N<m) (inverse e/2)) } ; i<b = <CInheritedR (<WellDefined groupIsAbelian identLeft (orderRespectsAddition (<WellDefined reflexive (invIdent additiveGroup) (ringSwapNegatives (<WellDefined (symmetric (invTwice additiveGroup 0R)) (symmetric (invTwice additiveGroup e/2)) (halvePositive' e/2+e/2=e 0<ε)))) b)) }
+
+<CRelaxR' : {a : CauchyCompletion} {b : A} → a <C injection b → a <Cr b
+<CRelaxR' {a} {b} record { i = i ; a<i = record { e = ε ; 0<e = 0<ε ; N = N ; property = property } ; i<b = i<b } = record { e = ε ; 0<e = 0<ε ; N = N ; property = λ m N<m → <Transitive (property m N<m) (<CCollapsesR _ _ i<b) }
 
 <CIrreflexive : {a : CauchyCompletion} → a <C a → False
-<CIrreflexive {a} (bound , ((bA , (0<bA ,, (Na , prA))) ,, ((bB , (0<bB ,, (Nb , prB)))))) with prA (succ Na +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Na)))
+<CIrreflexive {a} record { i = bound ; a<i = record { e = bA ; 0<e = 0<bA ; N = Na ; property = prA } ; i<b = record { e = bB ; 0<e = 0<bB ; N = Nb ; pr = prB } } with prA (succ Na +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Na)))
 ... | bl with prB (succ Na +N Nb) (le Na refl)
 ... | bl2 with <WellDefined (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) groupIsAbelian) (Equivalence.reflexive eq))) groupIsAbelian (ringAddInequalities bl bl2)
 ... | bad = irreflexive (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft +Associative (<WellDefined (Equivalence.reflexive eq) groupIsAbelian (orderRespectsAddition (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities 0<bA 0<bB)) (index (CauchyCompletion.elts a) (succ Na +N Nb) + bound)))) bad)
 
 <CTransitive : {a b c : CauchyCompletion} → a <C b → b <C c → a <C c
-<CTransitive {a} {b} {c} (boundA , ((eL , (0<eL ,, prL)) ,, (eR , (0<eR ,, (Nr , prR))))) (boundB , ((fL , (0<fL ,, (Nfl , prFl))) ,, (fR , (0<fR ,, (N , prFR))))) = boundA , ((eL , (0<eL ,, prL)) ,, (eR , (0<eR ,, (((Nr +N Nfl) +N N) , λ m N<m → SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (prR m (shrinkInequality {Nr} (shrinkInequality N<m))) (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<fL (index (CauchyCompletion.elts b) m)))) (prFl m (shrinkInequality' {Nr} (shrinkInequality N<m)))) (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<fR boundB))) (prFR m (shrinkInequality' N<m))))))
+<CTransitive {a} {b} {c} record { i = boundA ; a<i = record { e = eL ; 0<e = 0<eL ; N = NL ; property = prL } ; i<b = record { e = eR ; 0<e = 0<eR ; N = Nr ; pr = prR } } record { i = boundB ; a<i = record { e = fL ; 0<e = 0<fL ; N = Nfl ; property = prFl } ; i<b = record { e = fR ; 0<e = 0<fR ; N = N ; pr = prFR } } = record { i = boundA ; a<i = record { e = eL ; 0<e = 0<eL ; N = NL ; property = prL } ; i<b = record { e = eR ; 0<e = 0<eR ; N = (Nr +N Nfl) +N N ; pr = λ m N<m → SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (prR m (inequalityShrinkLeft {Nr} (inequalityShrinkLeft N<m))) (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<fL (index (CauchyCompletion.elts b) m)))) (prFl m (inequalityShrinkRight {Nr} (inequalityShrinkLeft N<m)))) (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<fR boundB))) (prFR m (inequalityShrinkRight N<m)) } }
 
 <COrder : SetoidPartialOrder cauchyCompletionSetoid _<C_
 SetoidPartialOrder.<WellDefined <COrder {a} {b} {c} {d} a=b c=d a<c = <CWellDefined {a} {b} {c} {d} a=b c=d a<c
diff --git a/Fields/CauchyCompletion/Definition.agda b/Fields/CauchyCompletion/Definition.agda
index dfb5bf1..2711ad0 100644
--- a/Fields/CauchyCompletion/Definition.agda
+++ b/Fields/CauchyCompletion/Definition.agda
@@ -42,3 +42,11 @@ CauchyCompletion.converges (injection a) = λ ε 0<e → 0 , λ {m} {n} _ _ →
   where
     t : (m n : ℕ) → index (constSequence a) m + inverse (index (constSequence a) n) ∼ 0R
     t m n = identityOfIndiscernablesLeft _∼_ (identityOfIndiscernablesLeft _∼_ invRight (equalityCommutative (applyEquality (λ i → a + inverse i) (indexAndConst a n)))) (applyEquality (_+ inverse (index (constSequence a) n)) (equalityCommutative (indexAndConst a m)))
+
+-- Some slightly odd things here relating to equality rather than equivalence. Ultimately this is here so we can say Q → R is a genuine injection, not just a setoid one.
+private
+  lemma : {x y : CauchyCompletion} → x ≡ y → CauchyCompletion.elts x ≡ CauchyCompletion.elts y
+  lemma {x} {.x} refl = refl
+
+CInjection' : Injection injection
+CInjection' pr = headInjective (lemma pr)
diff --git a/Fields/CauchyCompletion/Field.agda b/Fields/CauchyCompletion/Field.agda
index 47069be..f4739f5 100644
--- a/Fields/CauchyCompletion/Field.agda
+++ b/Fields/CauchyCompletion/Field.agda
@@ -4,7 +4,8 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import Setoids.Setoids
 open import Rings.Definition
 open import Rings.Lemmas
-open import Rings.Order
+open import Rings.Orders.Partial.Definition
+open import Rings.Orders.Total.Definition
 open import Groups.Definition
 open import Groups.Groups
 open import Fields.Fields
@@ -13,19 +14,20 @@ open import Sequences
 open import Setoids.Orders
 open import Functions
 open import LogicalFormulae
-open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
 
-module Fields.CauchyCompletion.Field {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder {_<_ = _<_} pOrder} {R : Ring S _+_ _*_} (order : OrderedRing R tOrder) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
+module Fields.CauchyCompletion.Field {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
 
 open Setoid S
-open SetoidTotalOrder tOrder
+open PartiallyOrderedRing pRing
 open SetoidPartialOrder pOrder
 open Equivalence eq
-open OrderedRing order
+open SetoidTotalOrder (TotallyOrderedRing.total order)
 open Field F
 open Group (Ring.additiveGroup R)
 
-open import Rings.Orders.Lemmas(order)
+open import Rings.Orders.Total.Lemmas order
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Multiplication order F charNot2
 open import Fields.CauchyCompletion.Addition order F charNot2
@@ -35,12 +37,12 @@ open import Fields.CauchyCompletion.Ring order F charNot2
 open import Fields.CauchyCompletion.Comparison order F charNot2
 
 Cnontrivial : (pr : Setoid._∼_ cauchyCompletionSetoid (injection (Ring.0R R)) (injection (Ring.1R R))) → False
-Cnontrivial pr with pr (Ring.1R R) (0<1 (charNot2ImpliesNontrivial charNot2))
+Cnontrivial pr with pr (Ring.1R R) (0<1 (charNot2ImpliesNontrivial R charNot2))
 Cnontrivial pr | N , b with b {succ N} (le 0 refl)
 ... | bl rewrite indexAndApply (constSequence 0G) (map inverse (constSequence (Ring.1R R))) _+_ {N} | indexAndConst 0G N | equalityCommutative (mapAndIndex (constSequence (Ring.1R R)) inverse N) | indexAndConst (Ring.1R R) N = irreflexive {Ring.1R R} (<WellDefined (Equivalence.transitive eq (absWellDefined _ _ identLeft) (Equivalence.transitive eq (Equivalence.symmetric eq (absNegation (Ring.1R R))) abs1Is1)) (Equivalence.reflexive eq) bl)
 
 boundedMap : A → A
-boundedMap a with SetoidTotalOrder.totality tOrder 0G a
+boundedMap a with totality 0G a
 boundedMap a | inl (inl x) = underlying (allInvertible a λ pr → irreflexive (<WellDefined (Equivalence.reflexive eq) pr x))
 boundedMap a | inl (inr x) = underlying (allInvertible a λ pr → irreflexive (<WellDefined pr (Equivalence.reflexive eq) x))
 boundedMap a | inr x = Ring.1R R
diff --git a/Fields/CauchyCompletion/Group.agda b/Fields/CauchyCompletion/Group.agda
index 26a2964..cb4ec48 100644
--- a/Fields/CauchyCompletion/Group.agda
+++ b/Fields/CauchyCompletion/Group.agda
@@ -25,32 +25,76 @@ open Equivalence eq
 open TotallyOrderedRing order
 open Field F
 open Group (Ring.additiveGroup R)
+open Ring R
 
 open import Rings.Orders.Total.Lemmas order
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Addition order F charNot2
 open import Fields.CauchyCompletion.Setoid order F charNot2
 
-Cassoc : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C (b +C c)) ((a +C b) +C c)
-Cassoc {a} {b} {c} ε 0<e = 0 , ans
-  where
-    ans : {m : ℕ} → 0 <N m → abs (index (CauchyCompletion.elts ((a +C (b +C c)) +C (-C ((a +C b) +C c)))) m) < ε
-    ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (b +C c))) (map inverse (CauchyCompletion.elts ((a +C b) +C c))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (apply _+_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (apply _+_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) +Associative)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
+abstract
+  +CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C b) (b +C a)
+  +CCommutative {a} {b} ε 0<e = 0 , ans
+    where
+      foo : {x y : A} → (x + y) + inverse (y + x) ∼ 0G
+      foo = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (inverseWellDefined additiveGroup groupIsAbelian)) invRight
+      ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a +C b)) (map inverse (CauchyCompletion.elts (b +C a)))) m) < ε
+      ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C b)) (map inverse (CauchyCompletion.elts (b +C a))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts a)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts a) _+_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ foo) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
 
-CidentRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C injection 0G) a
-CidentRight {a} ε 0<e = 0 , ans
-  where
-    ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a +C injection 0G)) (map inverse (CauchyCompletion.elts a))) m) < ε
-    ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C injection 0G)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (constSequence 0G) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined (identRight) (Equivalence.reflexive eq)) (invRight))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
+private
+  abstract
+    additionWellDefinedLeft : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid (a +C c) (b +C c)
+    additionWellDefinedLeft record { elts = a ; converges = aConv } record { elts = b ; converges = bConv } record { elts = c ; converges = cConv } a=b ε 0<e with a=b ε 0<e
+    ... | Na-b , prA-b = Na-b , ans
+      where
+        ans : {m : ℕ} → Na-b <N m → abs (index (apply _+_ (apply _+_ a c) (map inverse (apply _+_ b c))) m) < ε
+        ans {m} mBig with prA-b {m} mBig
+        ... | bl rewrite indexAndApply (apply _+_ a c) (map inverse (apply _+_ b c)) _+_ {m} | indexAndApply a c _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ b c) inverse m) | indexAndApply b c _+_ {m} = <WellDefined (absWellDefined _ _ t) (Equivalence.reflexive eq) bl
+          where
+            t : index (apply _+_ a (map inverse b)) m ∼ ((index a m + index c m) + inverse (index b m + index c m))
+            t rewrite indexAndApply a (map inverse b) _+_ {m} | equalityCommutative (mapAndIndex b inverse m) = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined (Equivalence.symmetric eq invRight) (Equivalence.reflexive eq))) (Equivalence.symmetric eq +Associative)) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (invContravariant additiveGroup))))) (+Associative {index a m})
 
-CidentLeft : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection 0G +C a) a
-CidentLeft {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection 0G +C a} {a +C injection 0G} {a} (+CCommutative {injection 0G} {a}) (CidentRight {a})
+    additionPreservedLeft : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (injection a +C c) (injection b +C c)
+    additionPreservedLeft {a} {b} {c} a=b = additionWellDefinedLeft (injection a) (injection b) c (injectionPreservesSetoid a b a=b)
 
-CinvRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C (-C a)) (injection 0G)
-CinvRight {a} ε 0<e = 0 , ans
-  where
-    ans : {m : ℕ} → (0 <N m) → abs (index (apply _+_ (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G)))) m) < ε
-    ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts a)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | equalityCommutative (mapAndIndex (constSequence 0G) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined invRight (invIdent (Ring.additiveGroup R))) identRight)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
+    additionPreservedRight : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (c +C injection a) (c +C injection b)
+    additionPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c +C injection a} {injection a +C c} {c +C injection b} (+CCommutative {c} {injection a}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C c} {injection b +C c} {c +C injection b} (additionPreservedLeft {a} {b} {c} a=b) (+CCommutative {injection b} {c}))
+
+    additionPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a +C injection c) (injection b +C injection d)
+    additionPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C injection c} {injection a +C injection d} {injection b +C injection d} (additionPreservedRight {c} {d} {injection a} c=d) (additionPreservedLeft {a} {b} {injection d} a=b)
+
+    additionWellDefinedRight : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid b c → Setoid._∼_ cauchyCompletionSetoid (a +C b) (a +C c)
+    additionWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C b} {b +C a} {a +C c} (+CCommutative {a} {b}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b +C a} {c +C a} {a +C c} (additionWellDefinedLeft b c a b=c) (+CCommutative {c} {a}))
+
+    additionWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a +C c) (b +C d)
+    additionWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C c} {a +C d} {b +C d} (additionWellDefinedRight a c d c=d) (additionWellDefinedLeft a b d a=b)
+
+    additionHom : (x y : A) → Setoid._∼_ cauchyCompletionSetoid (injection (x + y)) (injection x +C injection y)
+    additionHom x y ε 0<e = 0 , ans
+      where
+        ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (x + y))) (map inverse (CauchyCompletion.elts (injection x +C injection y)))) m) < ε
+        ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (injection (x + y))) (map inverse (CauchyCompletion.elts (injection x +C injection y))) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (constSequence x) (constSequence y)) inverse m) | indexAndConst (x + y) m | indexAndApply (constSequence x) (constSequence y) _+_ {m} | indexAndConst x m | indexAndConst y m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ invRight) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
+
+    Cassoc : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C (b +C c)) ((a +C b) +C c)
+    Cassoc {a} {b} {c} ε 0<e = 0 , ans
+      where
+        ans : {m : ℕ} → 0 <N m → abs (index (CauchyCompletion.elts ((a +C (b +C c)) +C (-C ((a +C b) +C c)))) m) < ε
+        ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (b +C c))) (map inverse (CauchyCompletion.elts ((a +C b) +C c))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (apply _+_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (apply _+_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) +Associative)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
+
+    CidentRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C injection 0G) a
+    CidentRight {a} ε 0<e = 0 , ans
+      where
+        ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a +C injection 0G)) (map inverse (CauchyCompletion.elts a))) m) < ε
+        ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C injection 0G)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (constSequence 0G) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined (identRight) (Equivalence.reflexive eq)) (invRight))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
+
+    CidentLeft : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection 0G +C a) a
+    CidentLeft {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection 0G +C a} {a +C injection 0G} {a} (+CCommutative {injection 0G} {a}) (CidentRight {a})
+
+    CinvRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C (-C a)) (injection 0G)
+    CinvRight {a} ε 0<e = 0 , ans
+      where
+        ans : {m : ℕ} → (0 <N m) → abs (index (apply _+_ (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G)))) m) < ε
+        ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts a)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | equalityCommutative (mapAndIndex (constSequence 0G) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined invRight (invIdent (Ring.additiveGroup R))) identRight)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
 
 CGroup : Group cauchyCompletionSetoid _+C_
 Group.+WellDefined CGroup {a} {b} {c} {d} x y = additionWellDefined {a} {c} {b} {d} x y
diff --git a/Fields/CauchyCompletion/Multiplication.agda b/Fields/CauchyCompletion/Multiplication.agda
index bb93a06..904ee5e 100644
--- a/Fields/CauchyCompletion/Multiplication.agda
+++ b/Fields/CauchyCompletion/Multiplication.agda
@@ -39,8 +39,9 @@ open import Fields.CauchyCompletion.Approximation order F charNot2
 0!=1 : {e : A} → (0G < e) → 0R ∼ 1R → False
 0!=1 {e} 0<e 0=1 = irreflexive (<WellDefined (Equivalence.reflexive eq) (oneZeroImpliesAllZero R 0=1) 0<e)
 
-littleLemma : {a b c d : A} → ((a * b) + inverse (c * d)) ∼ ((a * (b + inverse d)) + (d * (a + inverse c)))
-littleLemma {a} {b} {c} {d} = Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative)) (Equivalence.reflexive eq)) (invLeft {d * a}))) (Equivalence.transitive eq (Equivalence.symmetric eq (ringMinusExtracts' R)) *Commutative))) (Equivalence.symmetric eq +Associative))) (+Associative)) (Equivalence.symmetric eq (+WellDefined (*DistributesOver+) (*DistributesOver+)))
+private
+  littleLemma : {a b c d : A} → ((a * b) + inverse (c * d)) ∼ ((a * (b + inverse d)) + (d * (a + inverse c)))
+  littleLemma {a} {b} {c} {d} = Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative)) (Equivalence.reflexive eq)) (invLeft {d * a}))) (Equivalence.transitive eq (Equivalence.symmetric eq (ringMinusExtracts' R)) *Commutative))) (Equivalence.symmetric eq +Associative))) (+Associative)) (Equivalence.symmetric eq (+WellDefined (*DistributesOver+) (*DistributesOver+)))
 
 _*C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
 CauchyCompletion.elts (record { elts = a ; converges = aConv } *C record { elts = b ; converges = bConv }) = apply _*_ a b
@@ -99,7 +100,7 @@ CauchyCompletion.converges (record { elts = a ; converges = aConv } *C record {
     N : ℕ
     N = (Na +N (Nb +N (reallyNa +N reallyNb)))
     ans : {m : ℕ} {n : ℕ} → N <N m → N <N n → abs (index (apply _*_ a b) m + inverse (index (apply _*_ a b) n)) < e
-    ans {m} {n} N<m N<n rewrite indexAndApply a b _*_ {m} | indexAndApply a b _*_ {n} = ans'
+    ans {m} {n} N<m N<n rewrite indexAndApply a b _*_ {m} | indexAndApply a b _*_ {n} = ans'''
       where
         Na<m : Na <N m
         Na<m = inequalityShrinkLeft N<m
@@ -144,10 +145,10 @@ CauchyCompletion.converges (record { elts = a ; converges = aConv } *C record {
             ans'' with foo
             ... | inl pr = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar')) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar)) (orderRespectsAddition pr 0R))) q
             ... | inr pr = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.symmetric eq bar') (Equivalence.symmetric eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq pr)) absZeroIsZero)) timesZero)))) (Equivalence.reflexive eq) 0<e
-        ans' : abs ((index a m * index b m) + inverse (index a n * index b n)) < e
-        ans' with triangleInequality (index a m * (index b m + inverse (index b n))) (index b n * (index a m + inverse (index a n)))
+        ans''' : abs ((index a m * index b m) + inverse (index a n * index b n)) < e
+        ans''' with triangleInequality (index a m * (index b m + inverse (index b n))) (index b n * (index a m + inverse (index a n)))
         ... | inl less = <WellDefined (Equivalence.symmetric eq (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma)) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder less (<WellDefined (+WellDefined (Equivalence.symmetric eq (absRespectsTimes (index a m) _)) (Equivalence.symmetric eq (absRespectsTimes (index b n) _))) (Equivalence.reflexive eq) p))
-        ... | inr equal rewrite indexAndApply a b _*_ {m} | indexAndApply a b _*_ {n} = <WellDefined (Equivalence.symmetric eq (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma)) (Equivalence.reflexive eq) (<WellDefined (Equivalence.symmetric eq equal) (Equivalence.reflexive eq) ((<WellDefined (+WellDefined (Equivalence.symmetric eq (absRespectsTimes (index a m) _)) (Equivalence.symmetric eq (absRespectsTimes (index b n) _))) (Equivalence.reflexive eq) p)))
+        ... | inr equal = <WellDefined {_ + _} {abs _} {e} {e} (symmetric (transitive (transitive (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma) equal) (+WellDefined (absRespectsTimes (index a m) _) (absRespectsTimes (index b n) _)))) reflexive p
 
 *CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C b) (b *C a)
 *CCommutative {a} {b} ε 0<e = 0 , ans
@@ -162,17 +163,18 @@ abstract
   multiplicationWellDefinedLeft' : (0!=1 : 0R ∼ 1R → False) (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C c)
   multiplicationWellDefinedLeft' 0!=1 a b c a=b ε 0<e = N , ans
     where
-      cBoundAndPr : Sg A (λ b → Sg ℕ (λ N → (m : ℕ) → (N <N m) → (abs (index (CauchyCompletion.elts c) m)) < b))
-      cBoundAndPr = boundModulus c
-      cBound : A
-      cBound with cBoundAndPr
-      ... | a , _ = a
-      cBoundN : ℕ
-      cBoundN with cBoundAndPr
-      ... | _ , (N , _) = N
-      cBoundPr : (m : ℕ) → (cBoundN <N m) → (abs (index (CauchyCompletion.elts c) m)) < cBound
-      cBoundPr with cBoundAndPr
-      ... | _ , (_ , pr) = pr
+      abstract
+        cBoundAndPr : Sg A (λ b → Sg ℕ (λ N → (m : ℕ) → (N <N m) → (abs (index (CauchyCompletion.elts c) m)) < b))
+        cBoundAndPr = boundModulus c
+        cBound : A
+        cBound with cBoundAndPr
+        ... | a , _ = a
+        cBoundN : ℕ
+        cBoundN with cBoundAndPr
+        ... | _ , (N , _) = N
+        cBoundPr : (m : ℕ) → (cBoundN <N m) → (abs (index (CauchyCompletion.elts c) m)) < cBound
+        cBoundPr with cBoundAndPr
+        ... | _ , (_ , pr) = pr
       0<cBound : 0G < cBound
       0<cBound with totality 0G cBound
       0<cBound | inl (inl 0<cBound) = 0<cBound
@@ -197,10 +199,9 @@ abstract
       cBounded : (m : ℕ) → (N <N m) → abs (index (CauchyCompletion.elts c) m) < cBound
       cBounded m N<m = cBoundPr m (inequalityShrinkRight N<m)
       a-bSmall : (m : ℕ) → N <N m → abs ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts b) m)) < e/c
-      a-bSmall m N<m with abPr {m} (inequalityShrinkLeft N<m)
-      ... | f rewrite indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts b) inverse m) = f
+      a-bSmall m N<m = <WellDefined (absWellDefined _ _ (transitive (identityOfIndiscernablesLeft _∼_ reflexive (equalityCommutative (indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m}))) (+WellDefined reflexive (identityOfIndiscernablesLeft _∼_ reflexive (mapAndIndex (CauchyCompletion.elts b) inverse m))))) reflexive (abPr {m} (inequalityShrinkLeft N<m))
       ans : {m : ℕ} → N <N m → abs (index (apply _+_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) (map inverse (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)))) m) < ε
-      ans {m} N<m rewrite indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) (map inverse (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c))) _+_ {m} | equalityCommutative (mapAndIndex (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _*_ {m} = <WellDefined (absWellDefined _ _ (+WellDefined (Equivalence.reflexive eq) (ringMinusExtracts' R))) (Equivalence.reflexive eq) (<WellDefined (absWellDefined ((index (CauchyCompletion.elts a) m + inverse (index (CauchyCompletion.elts b) m)) * index (CauchyCompletion.elts c) m) _ (Equivalence.transitive eq (Equivalence.transitive eq *Commutative *DistributesOver+) (+WellDefined *Commutative *Commutative))) (Equivalence.reflexive eq) (<WellDefined (Equivalence.symmetric eq (absRespectsTimes _ _)) (Equivalence.reflexive eq) (<WellDefined (Equivalence.reflexive eq) e/cPr (ans' (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) (index (CauchyCompletion.elts c) m) (a-bSmall m N<m) (cBounded m N<m)))))
+      ans {m} N<m = <WellDefined (absWellDefined _ _ (transitive (+WellDefined (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply _ _ _*_ {m})) (transitive (transitive (ringMinusExtracts' R) (inverseWellDefined additiveGroup (identityOfIndiscernablesRight _∼_ reflexive (equalityCommutative (indexAndApply _ _ _*_ {m}))))) (identityOfIndiscernablesLeft _∼_ reflexive (equalityCommutative (mapAndIndex _ inverse m))))) (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply _ _ _+_ {m})))) reflexive (<WellDefined (absWellDefined ((index (CauchyCompletion.elts a) m + inverse (index (CauchyCompletion.elts b) m)) * index (CauchyCompletion.elts c) m) _ (Equivalence.transitive eq (Equivalence.transitive eq *Commutative *DistributesOver+) (+WellDefined *Commutative *Commutative))) (Equivalence.reflexive eq) (<WellDefined (Equivalence.symmetric eq (absRespectsTimes _ _)) (Equivalence.reflexive eq) (<WellDefined (Equivalence.reflexive eq) e/cPr (ans' (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) (index (CauchyCompletion.elts c) m) (a-bSmall m N<m) (cBounded m N<m)))))
         where
           ans' : (a b c : A) → abs (a + inverse b) < e/c → abs c < cBound → (abs (a + inverse b) * abs c) < (e/c * cBound)
           ans' a b c a-b<e/c c<bound with totality 0R c
diff --git a/Fields/CauchyCompletion/PartiallyOrderedRing.agda b/Fields/CauchyCompletion/PartiallyOrderedRing.agda
new file mode 100644
index 0000000..4e71343
--- /dev/null
+++ b/Fields/CauchyCompletion/PartiallyOrderedRing.agda
@@ -0,0 +1,160 @@
+{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import Setoids.Setoids
+open import Rings.Definition
+open import Rings.Lemmas
+open import Rings.Orders.Partial.Definition
+open import Rings.Orders.Total.Definition
+open import Groups.Definition
+open import Groups.Groups
+open import Fields.Fields
+open import Fields.Orders.Total.Definition
+open import Sets.EquivalenceRelations
+open import Sequences
+open import Setoids.Orders
+open import Functions
+open import LogicalFormulae
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
+open import Numbers.Naturals.Order.Lemmas
+open import Semirings.Definition
+open import Groups.Homomorphisms.Definition
+open import Rings.Homomorphisms.Definition
+
+module Fields.CauchyCompletion.PartiallyOrderedRing {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
+
+open Setoid S
+open SetoidTotalOrder (TotallyOrderedRing.total order)
+open SetoidPartialOrder pOrder
+open Equivalence eq
+open PartiallyOrderedRing pRing
+open Ring R
+open Group additiveGroup
+open Field F
+open import Fields.Lemmas F
+open import Rings.Orders.Partial.Lemmas pRing
+open import Rings.Orders.Total.Lemmas order
+open import Fields.Orders.Lemmas {F = F} {pRing} (record { oRing = order })
+open import Fields.CauchyCompletion.Definition order F
+open import Fields.CauchyCompletion.Addition order F charNot2
+open import Fields.CauchyCompletion.Multiplication order F charNot2
+open import Fields.CauchyCompletion.Approximation order F charNot2
+open import Fields.CauchyCompletion.Ring order F charNot2
+open import Fields.CauchyCompletion.Comparison order F charNot2
+open import Fields.CauchyCompletion.Setoid order F charNot2
+
+productPositives : (a b : A) → (injection 0R) <Cr a → (injection 0R) <Cr b → (injection 0R) <Cr (a * b)
+productPositives a b record { e = eA ; 0<e = 0<eA ; N = Na ; property = prA } record { e = eB ; 0<e = 0<eB ; N = Nb ; property = prB } = record { e = eA * eB ; 0<e = orderRespectsMultiplication 0<eA 0<eB ; N = Na +N Nb ; property = ans }
+  where
+    ans : (m : ℕ) → Na +N Nb <N m → ((eA * eB) + index (CauchyCompletion.elts (injection 0R)) m) < (a * b)
+    ans m <m = <WellDefined (symmetric (transitive (+WellDefined reflexive (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst 0R m))) identRight)) reflexive (ringMultiplyPositives 0<eA 0<eB (<WellDefined (transitive (+WellDefined reflexive (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst 0R m))) identRight) reflexive (prA m (inequalityShrinkLeft <m))) (<WellDefined (transitive (+WellDefined reflexive (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst 0R m))) identRight) reflexive (prB m (inequalityShrinkRight <m))))
+
+productPositives' : (a b : CauchyCompletion) (interA interB : A) → (0R < interA) → (0R < interB) → (interA r<C a) → (interB r<C b) → (interA * interB) r<C (a *C b)
+productPositives' a b interA interB 0<iA 0<iB record { e = interA' ; 0<e = 0<interA' ; N = Na ; pr = prA } record { e = interB' ; 0<e = 0<interB' ; N = Nb ; pr = prB } = record { e = interA' * interB' ; 0<e = orderRespectsMultiplication 0<interA' 0<interB' ; N = Na +N Nb ; pr = ans }
+  where
+    ans : (m : ℕ) → (Na +N Nb <N m) → ((interA * interB) + (interA' * interB')) < index (CauchyCompletion.elts (a *C b)) m
+    ans m <m rewrite indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} = <Transitive (<WellDefined identRight (symmetric *DistributesOver+) (<WellDefined reflexive (+WellDefined *Commutative *Commutative) (<WellDefined reflexive (+WellDefined (symmetric *DistributesOver+) (symmetric *DistributesOver+)) (<WellDefined groupIsAbelian (transitive (transitive groupIsAbelian (transitive (symmetric +Associative) (+WellDefined *Commutative (transitive groupIsAbelian (transitive (+WellDefined reflexive *Commutative) (symmetric +Associative)))))) +Associative) (orderRespectsAddition (<WellDefined identRight reflexive (ringAddInequalities (orderRespectsMultiplication 0<iB 0<interA') (orderRespectsMultiplication 0<interB' 0<iA))) ((interA * interB) + (interA' * interB'))))))) (ringMultiplyPositives (<WellDefined identRight reflexive (ringAddInequalities 0<iA 0<interA')) (<WellDefined identRight reflexive (ringAddInequalities 0<iB 0<interB')) (prA m (inequalityShrinkLeft <m)) (prB m (inequalityShrinkRight <m)))
+
+
+-- a  < a'
+-- b' < b
+-- then:
+-- a +C c < a' + c ~ a' + c' < b' + c' ~ b' + c < b +C c
+{-
+Have: a <Cr x r<C b
+
+* Let e = min(distance from a to witness of a<x, distance from x to witness of x<b)
+* Approximate a above to within e/2
+* Approximate b below to within e/2
+* Approximate c (above or below) to within e/2
+
+Then a' + c' is an appropriate witness.
+-}
+
+cOrderRespectsAdditionLeft' : (a : CauchyCompletion) (b : A) (c : A) → (a <Cr b) → (a +C injection c) <C (injection b +C injection c)
+cOrderRespectsAdditionLeft' a b c record { e = e ; 0<e = 0<e ; N = N ; property = pr } = <CWellDefined {a +C injection c} {a +C injection c} {injection (b + c)} {(injection b) +C (injection c)} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {a +C injection c}) (GroupHom.groupHom (RingHom.groupHom CInjectionRingHom)) (<CRelaxR (record { e = e ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive (ans m))) reflexive (orderRespectsAddition (pr m N<m) c) }))
+  where
+    ans : (m : ℕ) → (index (CauchyCompletion.elts a) m + c) ∼ index (apply _+_ (CauchyCompletion.elts a) (constSequence c)) m
+    ans m rewrite indexAndApply (CauchyCompletion.elts a) (constSequence c) _+_ {m} | indexAndConst c m = reflexive
+
+cOrderRespectsAdditionLeft : (a : CauchyCompletion) (b : A) (c : CauchyCompletion) → (a <Cr b) → (a +C c) <C (injection b +C c)
+cOrderRespectsAdditionLeft a b c a<b = {!!}
+
+cOrderRespectsAdditionRight : (a : A) (b : CauchyCompletion) (c : CauchyCompletion) → (a r<C b) → (injection a +C c) <C (b +C c)
+cOrderRespectsAdditionRight a b a<b = {!!}
+
+cOrderRespectsAddition : (a b : CauchyCompletion) → (a <C b) → (c : CauchyCompletion) → (a +C c) <C (b +C c)
+cOrderRespectsAddition a b a<b c = {!!}
+
+{-
+cOrderRespectsAddition : (a b : CauchyCompletion) → (a <C b) → (c : CauchyCompletion) → (a +C c) <C (b +C c)
+cOrderRespectsAddition a b (r , ((r1 , (0<r1 ,, (N1 , prN1))) ,, (r2 , (0<r2 ,, (N2 , prN2))))) c = (a' + c') , (ans1 ,, ans2)
+  where
+    0<min : 0G < min r1 r2
+    0<min with totality r1 r2
+    0<min | inl (inl r1<r2) = 0<r1
+    0<min | inl (inr r2<r1) = 0<r2
+    0<min | inr r1=r2 = 0<r1
+    e/2All : Sg A (λ i → i + i ∼ min r1 r2)
+    e/2All = halve charNot2 (min r1 r2)
+    e/2 : A
+    e/2 = underlying e/2All
+    prE/2 : e/2 + e/2 ∼ min r1 r2
+    prE/2 with e/2All
+    ... | _ , pr = pr
+    0<e/2 : 0G < e/2
+    0<e/2 = halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<min)
+    a'All : Sg A (λ i → (a <Cr i) && (injection i +C (-C a)) <C (injection e/2))
+    a' : A
+    a<a' : a <Cr a'
+    a'Pr : (injection a' +C (-C a)) <C (injection e/2)
+    b'All : Sg A (λ i → (i r<C b) && (b +C (-C injection i)) <C (injection e/2))
+    b' : A
+    b'<b : b' r<C b
+    b'Pr : (b +C (-C injection b')) <C (injection e/2)
+
+    c'All : Sg A (λ i → (c <Cr i) && (injection i +C (-C c)) <C (injection e/2))
+    c' : A
+    c<c' : c <Cr c'
+    c'Pr : (injection c' +C (-C c)) <C (injection e/2)
+
+    -- Now a' + c' is our intervening rational
+    -- and r1 suffices for the witness to a + c < a' + c'
+    -- and r2 suffices for the witness to b' + c' < b + c
+    -- TODO here
+
+    ans1 : (a +C c) <Cr (a' + c')
+    ans1 = r1 , (0<r1 ,, ((N1 +N N2) , ans))
+      where
+        ans : (m : ℕ) → (N1 +N N2) <N m → (r1 + index (CauchyCompletion.elts (a +C c)) m) < (a' + c')
+        ans m N1+N2<m rewrite indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _+_ {m} = <WellDefined (Equivalence.symmetric eq +Associative) reflexive (SetoidPartialOrder.<Transitive pOrder (orderRespectsAddition (prN1 m (inequalityShrinkLeft N1+N2<m)) (index (CauchyCompletion.elts c) m)) {!!})
+
+    ans2 : (a' + c') r<C (b +C c)
+    ans2 = r2 , (0<r2 ,, {!!})
+
+    a'All = approximateAbove a e/2 0<e/2
+    a' = underlying a'All
+    a<a' with a'All
+    ... | (_ , (x ,, _)) = x
+    a'Pr with a'All
+    ... | (_ , (_ ,, x)) = x
+    b'All = approximateBelow b e/2 0<e/2
+    b' = underlying b'All
+    b'<b with b'All
+    ... | (_ , (x ,, _)) = x
+    b'Pr with b'All
+    ... | (_ , (_ ,, x)) = x
+    c'All = approximateAbove c e/2 0<e/2
+    c' = underlying c'All
+    c<c' with c'All
+    ... | (_ , (x ,, _)) = x
+    c'Pr with c'All
+    ... | (_ , (_ ,, x)) = x
+
+-}
+
+CpOrderedRing : PartiallyOrderedRing CRing <COrder
+PartiallyOrderedRing.orderRespectsAddition CpOrderedRing {a} {b} = cOrderRespectsAddition a b
+PartiallyOrderedRing.orderRespectsMultiplication CpOrderedRing {a} {b} record { i = interA ; a<i = 0<interA ; i<b = interA<a } record { i = interB ; a<i = 0<interB ; i<b = interB<b } = record { i = interA * interB ; a<i = productPositives interA interB 0<interA 0<interB ; i<b = productPositives' a b interA interB (<CCollapsesL 0R _ 0<interA) (<CCollapsesL 0R _ 0<interB) interA<a interB<b }
+
diff --git a/Fields/CauchyCompletion/Ring.agda b/Fields/CauchyCompletion/Ring.agda
index 6b7fe57..04a9cdf 100644
--- a/Fields/CauchyCompletion/Ring.agda
+++ b/Fields/CauchyCompletion/Ring.agda
@@ -14,6 +14,7 @@ open import Functions
 open import LogicalFormulae
 open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Order
+open import Rings.Homomorphisms.Definition
 
 module Fields.CauchyCompletion.Ring {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
 
@@ -32,23 +33,25 @@ open import Fields.CauchyCompletion.Addition order F charNot2
 open import Fields.CauchyCompletion.Setoid order F charNot2
 open import Fields.CauchyCompletion.Group order F charNot2
 
-c*Assoc : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b *C c)) ((a *C b) *C c)
-c*Assoc {a} {b} {c} ε 0<e = 0 , ans
-  where
-    ans : {m : ℕ} → 0 <N m → abs (index (CauchyCompletion.elts ((a *C (b *C c)) +C (-C ((a *C b) *C c)))) m) < ε
-    ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b *C c))) (CauchyCompletion.elts (-C ((a *C b) *C c))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _*_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*Associative R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
+private
+  abstract
+    c*Assoc : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b *C c)) ((a *C b) *C c)
+    c*Assoc {a} {b} {c} ε 0<e = 0 , ans
+      where
+        ans : {m : ℕ} → 0 <N m → abs (index (CauchyCompletion.elts ((a *C (b *C c)) +C (-C ((a *C b) *C c)))) m) < ε
+        ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b *C c))) (CauchyCompletion.elts (-C ((a *C b) *C c))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _*_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*Associative R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
 
-c*Ident : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection (Ring.1R R) *C a) a
-c*Ident {a} ε 0<e = 0 , ans
-  where
-    ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a))) m) < ε
-    ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (constSequence (Ring.1R R)) (CauchyCompletion.elts a) _*_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst (Ring.1R R) m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.identIsIdent R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero)))) (Equivalence.reflexive eq) 0<e
+    c*Ident : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection (Ring.1R R) *C a) a
+    c*Ident {a} ε 0<e = 0 , ans
+      where
+        ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a))) m) < ε
+        ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (constSequence (Ring.1R R)) (CauchyCompletion.elts a) _*_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst (Ring.1R R) m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.identIsIdent R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero)))) (Equivalence.reflexive eq) 0<e
 
-*CDistribute : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b +C c)) ((a *C b) +C (a *C c))
-*CDistribute {a} {b} {c} e 0<e = 0 , ans
-  where
-    ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c))))) m) < e
-    ans {m} N<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c)))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _+_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c))) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*DistributesOver+ R))) (absZeroIsZero))) (Equivalence.reflexive eq) 0<e
+    *CDistribute : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b +C c)) ((a *C b) +C (a *C c))
+    *CDistribute {a} {b} {c} e 0<e = 0 , ans
+      where
+        ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c))))) m) < e
+        ans {m} N<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c)))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _+_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c))) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*DistributesOver+ R))) (absZeroIsZero))) (Equivalence.reflexive eq) 0<e
 
 CRing : Ring cauchyCompletionSetoid _+C_ _*C_
 Ring.additiveGroup CRing = CGroup
@@ -59,3 +62,15 @@ Ring.*Associative CRing {a} {b} {c} = c*Assoc {a} {b} {c}
 Ring.*Commutative CRing {a} {b} = *CCommutative {a} {b}
 Ring.*DistributesOver+ CRing {a} {b} {c} = *CDistribute {a} {b} {c}
 Ring.identIsIdent CRing {a} = c*Ident {a}
+
+private
+  injectionIsRingHom : (a b : A) → Setoid._∼_ cauchyCompletionSetoid (injection (a * b)) (injection a *C injection b)
+  injectionIsRingHom a b ε 0<e = 0 , ans
+    where
+      ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (a * b))) (map inverse (CauchyCompletion.elts (injection a *C injection b)))) m) < ε
+      ans {m} 0<m rewrite indexAndApply (constSequence (a * b)) (map inverse (apply _*_ (constSequence a) (constSequence b))) _+_ {m} | indexAndConst (a * b) m | equalityCommutative (mapAndIndex (apply _*_ (constSequence a) (constSequence b)) inverse m) | indexAndApply (constSequence a) (constSequence b) _*_ {m} | indexAndConst a m | indexAndConst b m = <WellDefined (symmetric (transitive (absWellDefined _ _ invRight) absZeroIsZero)) reflexive 0<e
+
+CInjectionRingHom : RingHom R CRing injection
+RingHom.preserves1 CInjectionRingHom = Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {injection (Ring.1R R)}
+RingHom.ringHom CInjectionRingHom {a} {b} = injectionIsRingHom a b
+RingHom.groupHom CInjectionRingHom = CInjectionGroupHom
diff --git a/Fields/CauchyCompletion/Setoid.agda b/Fields/CauchyCompletion/Setoid.agda
index 6a46a7a..ada26d2 100644
--- a/Fields/CauchyCompletion/Setoid.agda
+++ b/Fields/CauchyCompletion/Setoid.agda
@@ -91,46 +91,6 @@ injectionPreservesSetoid' a b a=b = transferToRight additiveGroup (absZeroImplie
     ... | inl x = x
     ... | inr x = exFalso (absNonnegative x)
 
-+CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C b) (b +C a)
-+CCommutative {a} {b} ε 0<e = 0 , ans
-  where
-    foo : {x y : A} → (x + y) + inverse (y + x) ∼ 0G
-    foo = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (inverseWellDefined additiveGroup groupIsAbelian)) invRight
-    ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a +C b)) (map inverse (CauchyCompletion.elts (b +C a)))) m) < ε
-    ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C b)) (map inverse (CauchyCompletion.elts (b +C a))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts a)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts a) _+_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ foo) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
-
-additionWellDefinedLeft : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid (a +C c) (b +C c)
-additionWellDefinedLeft record { elts = a ; converges = aConv } record { elts = b ; converges = bConv } record { elts = c ; converges = cConv } a=b ε 0<e with a=b ε 0<e
-... | Na-b , prA-b = Na-b , ans
-  where
-    ans : {m : ℕ} → Na-b <N m → abs (index (apply _+_ (apply _+_ a c) (map inverse (apply _+_ b c))) m) < ε
-    ans {m} mBig with prA-b {m} mBig
-    ... | bl rewrite indexAndApply (apply _+_ a c) (map inverse (apply _+_ b c)) _+_ {m} | indexAndApply a c _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ b c) inverse m) | indexAndApply b c _+_ {m} = <WellDefined (absWellDefined _ _ t) (Equivalence.reflexive eq) bl
-      where
-        t : index (apply _+_ a (map inverse b)) m ∼ ((index a m + index c m) + inverse (index b m + index c m))
-        t rewrite indexAndApply a (map inverse b) _+_ {m} | equalityCommutative (mapAndIndex b inverse m) = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined (Equivalence.symmetric eq invRight) (Equivalence.reflexive eq))) (Equivalence.symmetric eq +Associative)) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (invContravariant additiveGroup))))) (+Associative {index a m})
-
-additionPreservedLeft : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (injection a +C c) (injection b +C c)
-additionPreservedLeft {a} {b} {c} a=b = additionWellDefinedLeft (injection a) (injection b) c (injectionPreservesSetoid a b a=b)
-
-additionPreservedRight : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (c +C injection a) (c +C injection b)
-additionPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c +C injection a} {injection a +C c} {c +C injection b} (+CCommutative {c} {injection a}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C c} {injection b +C c} {c +C injection b} (additionPreservedLeft {a} {b} {c} a=b) (+CCommutative {injection b} {c}))
-
-additionPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a +C injection c) (injection b +C injection d)
-additionPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C injection c} {injection a +C injection d} {injection b +C injection d} (additionPreservedRight {c} {d} {injection a} c=d) (additionPreservedLeft {a} {b} {injection d} a=b)
-
-additionWellDefinedRight : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid b c → Setoid._∼_ cauchyCompletionSetoid (a +C b) (a +C c)
-additionWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C b} {b +C a} {a +C c} (+CCommutative {a} {b}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b +C a} {c +C a} {a +C c} (additionWellDefinedLeft b c a b=c) (+CCommutative {c} {a}))
-
-additionWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a +C c) (b +C d)
-additionWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C c} {a +C d} {b +C d} (additionWellDefinedRight a c d c=d) (additionWellDefinedLeft a b d a=b)
-
-additionHom : (x y : A) → Setoid._∼_ cauchyCompletionSetoid (injection (x + y)) (injection x +C injection y)
-additionHom x y ε 0<e = 0 , ans
-  where
-    ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (x + y))) (map inverse (CauchyCompletion.elts (injection x +C injection y)))) m) < ε
-    ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (injection (x + y))) (map inverse (CauchyCompletion.elts (injection x +C injection y))) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (constSequence x) (constSequence y)) inverse m) | indexAndConst (x + y) m | indexAndApply (constSequence x) (constSequence y) _+_ {m} | indexAndConst x m | indexAndConst y m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ invRight) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
-
 CInjection : SetoidInjection S cauchyCompletionSetoid injection
 SetoidInjection.wellDefined CInjection {x} {y} x=y = injectionPreservesSetoid x y x=y
 SetoidInjection.injective CInjection {x} {y} x=y = injectionPreservesSetoid' x y x=y
diff --git a/Fields/FieldOfFractions/Order.agda b/Fields/FieldOfFractions/Order.agda
index ed9e8ff..c589512 100644
--- a/Fields/FieldOfFractions/Order.agda
+++ b/Fields/FieldOfFractions/Order.agda
@@ -12,11 +12,11 @@ open import Setoids.Setoids
 open import Setoids.Orders
 open import Sets.EquivalenceRelations
 
-
 module Fields.FieldOfFractions.Order {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (I : IntegralDomain R) (order : TotallyOrderedRing pRing) where
 
 open import Fields.FieldOfFractions.Setoid I
 open import Fields.FieldOfFractions.Ring I
+open import Fields.FieldOfFractions.Addition I
 
 open SetoidTotalOrder (TotallyOrderedRing.total order)
 open import Rings.Orders.Partial.Lemmas
@@ -40,576 +40,594 @@ fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , den
   where
     open Equivalence (Setoid.eq S)
 
-fieldOfFractionsOrderWellDefinedLeft : {x y z : fieldOfFractionsSet} → fieldOfFractionsComparison x y → Setoid._∼_ fieldOfFractionsSetoid x z → fieldOfFractionsComparison z y
-fieldOfFractionsOrderWellDefinedLeft {(numX ,, (denomX , denomX!=0))} {(numY ,, (denomY , denomY!=0))} {(numZ ,, (denomZ , denomZ!=0))} x<y x=z with totality (Ring.0R R) denomZ
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = s
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
-    have = ringCanMultiplyByPositive pRing 0<denomZ x<y
-    p : ((numX * denomZ) * denomY) < ((numY * denomX) * denomZ)
-    p = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive have
-    q : ((denomX * numZ) * denomY) < ((numY * denomX) * denomZ)
-    q = SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=z reflexive) reflexive p
-    r : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
-    r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) *Commutative) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) q
-    s : (numZ * denomY) < (numY * denomZ)
-    s = ringCanCancelPositive order 0<denomX r
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (symmetric x))
-  where
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 r
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    p : ((numY * denomX) * denomZ) < ((numX * denomZ) * denomY)
-    p = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y)
-    q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
-    q = SetoidPartialOrder.<WellDefined pOrder reflexive (*WellDefined x=z reflexive) p
-    r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
-    r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) with totality (Ring.0R R) denomX
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX r
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    p : ((numY * denomX) * denomZ) < ((numX * denomY) * denomZ)
-    p = ringCanMultiplyByPositive pRing 0<denomZ x<y
-    q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
-    q = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined x=z reflexive)))) p
-    r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
-    r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 q
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
-    p = ringCanMultiplyByPositive pRing 0<denomZ x<y
-    q : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
-    q = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Commutative (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive (*Commutative)) (transitive *Associative (*WellDefined *Commutative reflexive))))) p)
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    p : ((numY * denomX) * denomZ) < ((denomY * numX) * denomZ)
-    p = ringCanMultiplyByNegative pRing denomZ<0 (SetoidPartialOrder.<WellDefined pOrder *Commutative reflexive x<y)
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined x=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p)
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
-    p = ringCanMultiplyByNegative pRing denomZ<0 x<y
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder denomY<0 0<denomY))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inr x = exFalso (denomX!=0 (symmetric x))
-  where
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) with totality (Ring.0R R) denomX
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Associative (transitive *Commutative *Associative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined *Commutative reflexive)))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (symmetric x))
-  where
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inr 0=denomZ = exFalso (denomZ!=0 (symmetric 0=denomZ))
-  where
-    open Equivalence (Setoid.eq S)
+private
+  abstract
 
-fieldOfFractionsOrderWellDefinedRight : {x y z : fieldOfFractionsSet} → fieldOfFractionsComparison x y → Setoid._∼_ (fieldOfFractionsSetoid) y z → fieldOfFractionsComparison x z
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z with totality (Ring.0R R) denomX
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) with totality (Ring.0R R) denomZ
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive (*Associative) (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) with totality (Ring.0R R) denomZ
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
-  where
-    open Ring R
-    open Equivalence (Setoid.eq S)
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
-fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft : {x y z : fieldOfFractionsSet} → fieldOfFractionsComparison x y → Setoid._∼_ fieldOfFractionsSetoid x z → fieldOfFractionsComparison z y
+    fieldOfFractionsOrderWellDefinedLeft {(numX ,, (denomX , denomX!=0))} {(numY ,, (denomY , denomY!=0))} {(numZ ,, (denomZ , denomZ!=0))} x<y x=z with totality (Ring.0R R) denomZ
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = s
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
+        have = ringCanMultiplyByPositive pRing 0<denomZ x<y
+        p : ((numX * denomZ) * denomY) < ((numY * denomX) * denomZ)
+        p = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive have
+        q : ((denomX * numZ) * denomY) < ((numY * denomX) * denomZ)
+        q = SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=z reflexive) reflexive p
+        r : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
+        r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) *Commutative) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) q
+        s : (numZ * denomY) < (numY * denomZ)
+        s = ringCanCancelPositive order 0<denomX r
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (symmetric x))
+      where
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 r
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        p : ((numY * denomX) * denomZ) < ((numX * denomZ) * denomY)
+        p = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y)
+        q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
+        q = SetoidPartialOrder.<WellDefined pOrder reflexive (*WellDefined x=z reflexive) p
+        r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
+        r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) with totality (Ring.0R R) denomX
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX r
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        p : ((numY * denomX) * denomZ) < ((numX * denomY) * denomZ)
+        p = ringCanMultiplyByPositive pRing 0<denomZ x<y
+        q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
+        q = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined x=z reflexive)))) p
+        r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
+        r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 q
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
+        p = ringCanMultiplyByPositive pRing 0<denomZ x<y
+        q : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
+        q = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Commutative (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) with totality (Ring.0R R) denomX
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive (*Commutative)) (transitive *Associative (*WellDefined *Commutative reflexive))))) p)
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        p : ((numY * denomX) * denomZ) < ((denomY * numX) * denomZ)
+        p = ringCanMultiplyByNegative pRing denomZ<0 (SetoidPartialOrder.<WellDefined pOrder *Commutative reflexive x<y)
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined x=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p)
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
+        p = ringCanMultiplyByNegative pRing denomZ<0 x<y
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder denomY<0 0<denomY))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inr x = exFalso (denomX!=0 (symmetric x))
+      where
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) with totality (Ring.0R R) denomX
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Associative (transitive *Commutative *Associative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined *Commutative reflexive)))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inr 0=denomZ = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomZ))
 
-swapLemma : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) {x y z : A} → Setoid._∼_ S ((x * y) * z) ((x * z) * y)
-swapLemma {S = S} R = transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight : {x y z : fieldOfFractionsSet} → fieldOfFractionsComparison x y → Setoid._∼_ (fieldOfFractionsSetoid) y z → fieldOfFractionsComparison x z
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z with totality (Ring.0R R) denomX
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) with totality (Ring.0R R) denomZ
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive (*Associative) (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) with totality (Ring.0R R) denomZ
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
+      where
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
+
+    swapLemma : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) {x y z : A} → Setoid._∼_ S ((x * y) * z) ((x * z) * y)
+    swapLemma {S = S} R = transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+
+private
+  abstract
+    irreflexive : (a : fieldOfFractionsSet) (pr : fieldOfFractionsComparison a a) → False
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr with totality (Ring.0R R) aDenom
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) with totality (Ring.0R R) aDenom
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) | inl (inl _) = SetoidPartialOrder.irreflexive pOrder pr
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) | inl (inr aDenom<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<aDenom aDenom<0))
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inl 0<aDenom) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) with totality (Ring.0R R) aDenom
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) | inl (inl 0<aDenom) = SetoidPartialOrder.irreflexive pOrder pr
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) | inl (inr _) = SetoidPartialOrder.irreflexive pOrder pr
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inl (inr aDenom<0) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    irreflexive (aNum ,, (aDenom , aDenom!=0)) pr | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
+
+    <transitive : (a b c : fieldOfFractionsSet) (a<b : fieldOfFractionsComparison a b) (b<c : fieldOfFractionsComparison b c) → fieldOfFractionsComparison a c
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c with totality (Ring.0R R) denomA
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl x) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB p
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
+        inter = ringCanMultiplyByPositive pRing 0<denomC a<b
+        p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
+        p = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive inter) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomA b<c))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomC a<b)))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) with totality (Ring.0R R) denomB
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomC<0 a<b)))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        have : ((numC * denomA) * denomB) < ((numB * denomC)  * denomA)
+        have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
+    ... | (inl (inl 0<denomC)) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    ... | (inl (inr _)) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive pRing 0<denomA b<c)))
+          where
+            open Setoid S
+            open Ring R
+            open Equivalence (Setoid.eq S)
+            have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
+            have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
+    ... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
+        have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByPositive pRing 0<denomC a<b)
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
+        have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive pRing 0<denomC a<b)
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with totality (Ring.0R R) denomB
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
+        have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+        have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
+        have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByNegative pRing denomC<0 a<b)
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+    <transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
 
 fieldOfFractionsOrder : SetoidPartialOrder fieldOfFractionsSetoid fieldOfFractionsComparison
 SetoidPartialOrder.<WellDefined (fieldOfFractionsOrder) {a} {b} {c} {d} a=b c=d a<c = fieldOfFractionsOrderWellDefinedRight {b} {c} {d} (fieldOfFractionsOrderWellDefinedLeft {a} {c} {b} a<c a=b) c=d
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr with totality (Ring.0R R) aDenom
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inl 0<aDenom) with totality (Ring.0R R) aDenom
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inl 0<aDenom) | inl (inl _) = SetoidPartialOrder.irreflexive pOrder pr
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inl 0<aDenom) | inl (inr aDenom<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<aDenom aDenom<0))
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inl 0<aDenom) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inr aDenom<0) with totality (Ring.0R R) aDenom
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inr aDenom<0) | inl (inl 0<aDenom) = SetoidPartialOrder.irreflexive pOrder pr
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inr aDenom<0) | inl (inr _) = SetoidPartialOrder.irreflexive pOrder pr
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inr aDenom<0) | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {aNum ,, (aDenom , aDenom!=0)} pr | inr x = exFalso (aDenom!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c with totality (Ring.0R R) denomA
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl x) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB p
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
-    inter = ringCanMultiplyByPositive pRing 0<denomC a<b
-    p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
-    p = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive inter) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomA b<c))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomC a<b)))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) with totality (Ring.0R R) denomB
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomC<0 a<b)))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    have : ((numC * denomA) * denomB) < ((numB * denomC)  * denomA)
-    have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
-... | (inl (inl 0<denomC)) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-... | (inl (inr _)) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive pRing 0<denomA b<c)))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
-    have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
-... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
-    have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByPositive pRing 0<denomC a<b)
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
-    have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive pRing 0<denomC a<b)
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with totality (Ring.0R R) denomB
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
-    have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-    have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
-    have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByNegative pRing denomC<0 a<b)
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+SetoidPartialOrder.irreflexive (fieldOfFractionsOrder) {a} pr = irreflexive a pr
+SetoidPartialOrder.<Transitive (fieldOfFractionsOrder) {a} {b} {c} a<b b<c = <transitive a b c a<b b<c
+
+private
+  <totality : (a b : fieldOfFractionsSet) → ((fieldOfFractionsComparison a b) || (fieldOfFractionsComparison b a)) || (Setoid._∼_ fieldOfFractionsSetoid a b)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) with totality (Ring.0R R) denomA
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) with totality (numA * denomB) (numB * denomA)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inl x) = inl (inl x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inr x) = inl (inr x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inr x = inr (Equivalence.transitive (Setoid.eq S) x (Ring.*Commutative R))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inr denomA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) with totality (Ring.0R R) denomA
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) with totality (numB * denomA) (numA * denomB)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inl x) = inl (inl x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inr x) = inl (inr x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inr x = inr (Equivalence.symmetric (Setoid.eq S) (Equivalence.transitive (Setoid.eq S) (Ring.*Commutative R) x))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) with totality (Ring.0R R) denomB
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) with totality (numB * denomA) (numA * denomB)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inl x) = inl (inl x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inr x) = inl (inr x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inr x = inr (Equivalence.transitive (Setoid.eq S) (Equivalence.symmetric (Setoid.eq S) x) (Ring.*Commutative R))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inl 0<denomA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) with totality (numA * denomB) (numB * denomA)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inl x) = inl (inl x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inr x) = inl (inr x)
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inr x = inr (Equivalence.transitive (Setoid.eq S) x (Ring.*Commutative R))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <totality (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
 
 fieldOfFractionsTotalOrder : SetoidTotalOrder fieldOfFractionsOrder
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) with totality (Ring.0R R) denomA
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) with totality (numA * denomB) (numB * denomA)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inl x) = inl (inl x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inl (inr x) = inl (inr x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inl _) | inr x = inr (Equivalence.transitive (Setoid.eq S) x (Ring.*Commutative R))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inl (inr denomA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) with totality (Ring.0R R) denomA
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) with totality (numB * denomA) (numA * denomB)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inl x) = inl (inl x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inl (inr x) = inl (inr x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inl _) | inr x = inr (Equivalence.symmetric (Setoid.eq S) (Equivalence.transitive (Setoid.eq S) (Ring.*Commutative R) x))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) with totality (Ring.0R R) denomB
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) with totality (numB * denomA) (numA * denomB)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inl x) = inl (inl x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inl (inr x) = inl (inr x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inl (inr _) | inr x = inr (Equivalence.transitive (Setoid.eq S) (Equivalence.symmetric (Setoid.eq S) x) (Ring.*Commutative R))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inl 0<denomA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomA denomA<0))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) with totality (numA * denomB) (numB * denomA)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inl x) = inl (inl x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inl (inr x) = inl (inr x)
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inl (inr _) | inr x = inr (Equivalence.transitive (Setoid.eq S) x (Ring.*Commutative R))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+SetoidTotalOrder.totality (fieldOfFractionsTotalOrder) x y = <totality x y
 
-ineqLemma : {x y : A} → (Ring.0R R) < (x * y) → (Ring.0R R) < x → (Ring.0R R) < y
-ineqLemma {x} {y} 0<xy 0<x with totality (Ring.0R R) y
-ineqLemma {x} {y} 0<xy 0<x | inl (inl 0<y) = 0<y
-ineqLemma {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing y<0 0<x))))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-ineqLemma {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
+private
+  abstract
 
-ineqLemma' : {x y : A} → (Ring.0R R) < (x * y) → x < (Ring.0R R) → y < (Ring.0R R)
-ineqLemma' {x} {y} 0<xy x<0 with totality (Ring.0R R) y
-... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing x<0 0<y))))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-... | inl (inr y<0) = y<0
-... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
+    ineqLemma : {x y : A} → (Ring.0R R) < (x * y) → (Ring.0R R) < x → (Ring.0R R) < y
+    ineqLemma {x} {y} 0<xy 0<x with totality (Ring.0R R) y
+    ineqLemma {x} {y} 0<xy 0<x | inl (inl 0<y) = 0<y
+    ineqLemma {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing y<0 0<x))))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    ineqLemma {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
 
-ineqLemma'' : {x y : A} → (x * y) < (Ring.0R R) → (Ring.0R R) < x → y < (Ring.0R R)
-ineqLemma'' {x} {y} xy<0 0<x with totality (Ring.0R R) y
-... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (orderRespectsMultiplication 0<x 0<y)))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-... | inl (inr y<0) = y<0
-... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
+    ineqLemma' : {x y : A} → (Ring.0R R) < (x * y) → x < (Ring.0R R) → y < (Ring.0R R)
+    ineqLemma' {x} {y} 0<xy x<0 with totality (Ring.0R R) y
+    ... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing x<0 0<y))))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    ... | inl (inr y<0) = y<0
+    ... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
 
-ineqLemma''' : {x y : A} → (x * y) < (Ring.0R R) → x < (Ring.0R R) → (Ring.0R R) < y
-ineqLemma''' {x} {y} xy<0 x<0 with totality (Ring.0R R) y
-... | inl (inl 0<y) = 0<y
-... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing y<0 x<0))))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
-... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
-  where
-    open Setoid S
-    open Ring R
-    open Equivalence (Setoid.eq S)
+    ineqLemma'' : {x y : A} → (x * y) < (Ring.0R R) → (Ring.0R R) < x → y < (Ring.0R R)
+    ineqLemma'' {x} {y} xy<0 0<x with totality (Ring.0R R) y
+    ... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (orderRespectsMultiplication 0<x 0<y)))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    ... | inl (inr y<0) = y<0
+    ... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+
+    ineqLemma''' : {x y : A} → (x * y) < (Ring.0R R) → x < (Ring.0R R) → (Ring.0R R) < y
+    ineqLemma''' {x} {y} xy<0 x<0 with totality (Ring.0R R) y
+    ... | inl (inl 0<y) = 0<y
+    ... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing y<0 x<0))))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+    ... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
+      where
+        open Setoid S
+        open Ring R
+        open Equivalence (Setoid.eq S)
+
+private
+  <orderRespectsAddition : (a b : fieldOfFractionsSet) (a<b : fieldOfFractionsComparison a b) (c : fieldOfFractionsSet) → fieldOfFractionsComparison (fieldOfFractionsPlus a c) (fieldOfFractionsPlus b c)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) with totality (Ring.0R R) (denomA * denomC)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) with totality (Ring.0R R) (denomB * denomC)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) ans))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        0<dC : 0R < denomC
+        0<dC with totality 0R denomC
+        0<dC | inl (inl x) = x
+        0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dB))))
+        0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+        p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
+        p = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive pRing 0<dC a<b)
+        ans : ((((numA * denomC) * denomB) + ((denomA * numC) * denomB))) < ((((numB * denomC) * denomA) + ((denomB * numC) * denomA)))
+        ans = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (*WellDefined *Commutative reflexive) (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (*WellDefined *Commutative reflexive)))) (PartiallyOrderedRing.orderRespectsAddition pRing p ((denomA * numC) * denomB))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso bad
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 with totality 0R denomC
+        ... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive pRing x dB<0))))
+        ... | inl (inr x) = x
+        ... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+        bad : False
+        bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dA)))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = exFalso bad
+        where
+          open Setoid S
+          open Equivalence (Setoid.eq S)
+          open Ring R
+          0<dC : 0R < denomC
+          0<dC with totality 0R denomC
+          0<dC | inl (inl x) = x
+          0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dB))))
+          0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+          dC<0 : denomC < 0R
+          dC<0 with totality 0R denomC
+          dC<0 | inl (inl 0<dC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dA<0 0<dC))))
+          dC<0 | inl (inr x) = x
+          dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
+          bad : False
+          bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma' 0<dAdC dA<0
+        have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
+        have = ringCanMultiplyByNegative pRing dC<0 a<b
+        have' : (denomA * (numB * denomC)) < (denomB * (numA * denomC))
+        have' = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) have
+        have'' : ((denomA * (numB * denomC)) + (denomA * (denomB * numC))) < ((denomB * (numA * denomC)) + (denomB * (denomA * numC)))
+        have'' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (PartiallyOrderedRing.orderRespectsAddition pRing have' (denomA * (denomB * numC)))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso bad
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        0<dC : 0R < denomC
+        0<dC = ineqLemma 0<dAdC 0<dA
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma'' dBdC<0 0<dB
+        bad : False
+        bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC ans)
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        0<dC : 0R < denomC
+        0<dC = ineqLemma 0<dAdC 0<dA
+        have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
+        have = ringCanMultiplyByPositive pRing 0<dC a<b
+        have' : (((numB * denomC) * denomA) + ((denomB * numC) * denomA)) < (((numA * denomC) * denomB) + ((denomB * numC) * denomA))
+        have' = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) have) _
+        ans : (((numB * denomC) + (denomB * numC)) * denomA) < (((numA * denomC) + (denomA * numC)) * denomB)
+        ans = SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have'
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) have))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma'' dBdC<0 0<dB
+        have : (((numA * denomC) * denomB) + ((denomB * numC) * denomA)) < (((numB * denomC) * denomA) + ((denomB * numC) * denomA))
+        have = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing dC<0 a<b)) _
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = exFalso bad
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma' 0<dAdC dA<0
+        0<dC : 0R < denomC
+        0<dC = ineqLemma''' dBdC<0 dB<0
+        bad : False
+        bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inr 0=dBdC with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dBdC)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inr 0=dBdC | f = exFalso (denomC!=0 (f denomB!=0))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) with totality (Ring.0R R) (denomB * denomC)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        0<dC : 0R < denomC
+        0<dC = ineqLemma 0<dBdC 0<dB
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma'' dAdC<0 0<dA
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma'' dAdC<0 0<dA
+        have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
+        have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByNegative pRing dC<0 a<b) _
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr 0=dB = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) 0=dB))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) *Commutative)) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (symmetric *DistributesOver+)) *Commutative)) have))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        0<dC : 0R < denomC
+        0<dC = ineqLemma 0<dBdC 0<dB
+        have : (((numB * denomA) * denomC) + ((denomA * numC) * denomB)) < (((numA * denomB) * denomC) + ((denomA * numC) * denomB))
+        have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByPositive pRing 0<dC a<b) _
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma' 0<dBdC dB<0
+        0<dC : 0R < denomC
+        0<dC = ineqLemma''' dAdC<0 dA<0
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inr 0=dB = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) 0=dB))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inr 0=dA = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) 0=dA))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive)) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive (transitive *Associative (*WellDefined *Commutative reflexive)) *Commutative) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma'' dAdC<0 0<dA
+        have : (((numB * denomA) * denomC) + ((denomB * numC) * denomA)) < (((numA * denomB) * denomC) + ((denomB * numC) * denomA))
+        have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByNegative pRing dC<0 a<b) _
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dC<0 0<dC))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma'' dAdC<0 0<dA
+        0<dC : 0R < denomC
+        0<dC = ineqLemma''' dBdC<0 dB<0
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        0<dC : 0R < denomC
+        0<dC = ineqLemma''' dAdC<0 dA<0
+        dC<0 : denomC < 0R
+        dC<0 = ineqLemma'' dBdC<0 0<dB
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (transitive (symmetric *DistributesOver+) *Commutative))) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) (transitive *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)))) (symmetric *DistributesOver+)) *Commutative)) have))
+      where
+        open Setoid S
+        open Equivalence (Setoid.eq S)
+        open Ring R
+        0<dC : 0R < denomC
+        0<dC = ineqLemma''' dAdC<0 dA<0
+        have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
+        have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByPositive pRing 0<dC a<b) _
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inr 0=dBdC with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dBdC)
+  ... | f = exFalso (denomC!=0 (f denomB!=0))
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inr (0=dAdC) with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdC)
+  <orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inr 0=dAdC | f = exFalso (denomC!=0 (f denomA!=0))
 
 fieldOfFractionsPOrderedRing : PartiallyOrderedRing fieldOfFractionsRing (SetoidTotalOrder.partial fieldOfFractionsTotalOrder)
-PartiallyOrderedRing.orderRespectsAddition fieldOfFractionsPOrderedRing {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) with totality (Ring.0R R) (denomA * denomC)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) with totality (Ring.0R R) (denomB * denomC)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) ans))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC with totality 0R denomC
-    0<dC | inl (inl x) = x
-    0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dB))))
-    0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-    p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
-    p = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive pRing 0<dC a<b)
-    ans : ((((numA * denomC) * denomB) + ((denomA * numC) * denomB))) < ((((numB * denomC) * denomA) + ((denomB * numC) * denomA)))
-    ans = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (*WellDefined *Commutative reflexive) (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (*WellDefined *Commutative reflexive)))) (PartiallyOrderedRing.orderRespectsAddition pRing p ((denomA * numC) * denomB))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso bad
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 with totality 0R denomC
-    ... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive pRing x dB<0))))
-    ... | inl (inr x) = x
-    ... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-    bad : False
-    bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dA)))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = exFalso bad
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC with totality 0R denomC
-    0<dC | inl (inl x) = x
-    0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dB))))
-    0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-    dC<0 : denomC < 0R
-    dC<0 with totality 0R denomC
-    dC<0 | inl (inl 0<dC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dA<0 0<dC))))
-    dC<0 | inl (inr x) = x
-    dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
-    bad : False
-    bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma' 0<dAdC dA<0
-    have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
-    have = ringCanMultiplyByNegative pRing dC<0 a<b
-    have' : (denomA * (numB * denomC)) < (denomB * (numA * denomC))
-    have' = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) have
-    have'' : ((denomA * (numB * denomC)) + (denomA * (denomB * numC))) < ((denomB * (numA * denomC)) + (denomB * (denomA * numC)))
-    have'' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (PartiallyOrderedRing.orderRespectsAddition pRing have' (denomA * (denomB * numC)))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso bad
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC = ineqLemma 0<dAdC 0<dA
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma'' dBdC<0 0<dB
-    bad : False
-    bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC ans)
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC = ineqLemma 0<dAdC 0<dA
-    have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
-    have = ringCanMultiplyByPositive pRing 0<dC a<b
-    have' : (((numB * denomC) * denomA) + ((denomB * numC) * denomA)) < (((numA * denomC) * denomB) + ((denomB * numC) * denomA))
-    have' = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) have) _
-    ans : (((numB * denomC) + (denomB * numC)) * denomA) < (((numA * denomC) + (denomA * numC)) * denomB)
-    ans = SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have'
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) have))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma'' dBdC<0 0<dB
-    have : (((numA * denomC) * denomB) + ((denomB * numC) * denomA)) < (((numB * denomC) * denomA) + ((denomB * numC) * denomA))
-    have = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing dC<0 a<b)) _
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = exFalso bad
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma' 0<dAdC dA<0
-    0<dC : 0R < denomC
-    0<dC = ineqLemma''' dBdC<0 dB<0
-    bad : False
-    bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inr 0=dBdC with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dBdC)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inr 0=dBdC | f = exFalso (denomC!=0 (f denomB!=0))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) with totality (Ring.0R R) (denomB * denomC)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC = ineqLemma 0<dBdC 0<dB
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma'' dAdC<0 0<dA
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma'' dAdC<0 0<dA
-    have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
-    have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByNegative pRing dC<0 a<b) _
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr 0=dB = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) 0=dB))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) *Commutative)) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (symmetric *DistributesOver+)) *Commutative)) have))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC = ineqLemma 0<dBdC 0<dB
-    have : (((numB * denomA) * denomC) + ((denomA * numC) * denomB)) < (((numA * denomB) * denomC) + ((denomA * numC) * denomB))
-    have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByPositive pRing 0<dC a<b) _
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma' 0<dBdC dB<0
-    0<dC : 0R < denomC
-    0<dC = ineqLemma''' dAdC<0 dA<0
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inr 0=dB = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) 0=dB))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inr 0=dA = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) 0=dA))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) with totality (Ring.0R R) denomA
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive)) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive (transitive *Associative (*WellDefined *Commutative reflexive)) *Commutative) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma'' dAdC<0 0<dA
-    have : (((numB * denomA) * denomC) + ((denomB * numC) * denomA)) < (((numA * denomB) * denomC) + ((denomB * numC) * denomA))
-    have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByNegative pRing dC<0 a<b) _
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dC<0 0<dC))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma'' dAdC<0 0<dA
-    0<dC : 0R < denomC
-    0<dC = ineqLemma''' dBdC<0 dB<0
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC = ineqLemma''' dAdC<0 dA<0
-    dC<0 : denomC < 0R
-    dC<0 = ineqLemma'' dBdC<0 0<dB
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (transitive (symmetric *DistributesOver+) *Commutative))) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) (transitive *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)))) (symmetric *DistributesOver+)) *Commutative)) have))
-  where
-    open Setoid S
-    open Equivalence (Setoid.eq S)
-    open Ring R
-    0<dC : 0R < denomC
-    0<dC = ineqLemma''' dAdC<0 dA<0
-    have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
-    have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByPositive pRing 0<dC a<b) _
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inr 0=dBdC with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dBdC)
-... | f = exFalso (denomC!=0 (f denomB!=0))
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inr (0=dAdC) with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdC)
-PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inr 0=dAdC | f = exFalso (denomC!=0 (f denomA!=0))
+PartiallyOrderedRing.orderRespectsAddition fieldOfFractionsPOrderedRing {a} {b} a<b c = <orderRespectsAddition a b a<b c
 PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} t u with totality (Ring.0R R) (Ring.1R R)
 PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) with totality (Ring.0R R) (denomA * denomB)
 PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) with totality (Ring.0R R) denomB
diff --git a/LogicalFormulae.agda b/LogicalFormulae.agda
index 4795a8c..5a70556 100644
--- a/LogicalFormulae.agda
+++ b/LogicalFormulae.agda
@@ -32,6 +32,12 @@ record _&&_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
     fst : A
     snd : B
 
+equalityLeft : {a b : _} {A : Set a} {B : Set b} {x1 x2 : A && B} → (x1 ≡ x2) → _&&_.fst x1 ≡ _&&_.fst x2
+equalityLeft {x1 = a ,, _} {.a ,, _} refl = refl
+
+equalityRight : {a b : _} {A : Set a} {B : Set b} {x1 x2 : A && B} → (x1 ≡ x2) → _&&_.snd x1 ≡ _&&_.snd x2
+equalityRight {x1 = _ ,, a} {_ ,, .a} refl = refl
+
 infix 15 _&_&_
 record _&_&_ {a b c} (A : Set a) (B : Set b) (C : Set c) : Set (a ⊔ b ⊔ c) where
   field
@@ -77,9 +83,9 @@ orIsAssociative (inl (inl x)) = inl x
 orIsAssociative (inl (inr x)) = inr (inl x)
 orIsAssociative (inr x) = inr (inr x)
 
-leqConstructive : {n : _} → {p : Set n} → (p || (p → False)) → (((p → False) → False) → p)
-leqConstructive (inl p) = λ _ → p
-leqConstructive (inr notP) = λ negnegP → exFalso (negnegP notP)
+lemConstructive : {n : _} → {p : Set n} → (p || (p → False)) → (((p → False) → False) → p)
+lemConstructive (inl p) = λ _ → p
+lemConstructive (inr notP) = λ negnegP → exFalso (negnegP notP)
 
 lemConverse : {n : _} → {p : Set n} → p → ((p → False) → False)
 lemConverse p = λ notP → notP p
diff --git a/Numbers/Rationals/Definition.agda b/Numbers/Rationals/Definition.agda
index b76fbdc..f18b1f9 100644
--- a/Numbers/Rationals/Definition.agda
+++ b/Numbers/Rationals/Definition.agda
@@ -1,4 +1,4 @@
-{-# OPTIONS --safe --warning=error #-}
+{-# OPTIONS --safe --warning=error --without-K #-}
 
 open import LogicalFormulae
 open import Numbers.Naturals.Naturals
@@ -10,7 +10,6 @@ open import Rings.Definition
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Partial.Definition
 open import Fields.Fields
-open import Numbers.Primes.PrimeNumbers
 open import Setoids.Setoids
 open import Setoids.Orders
 open import Functions
@@ -28,6 +27,9 @@ open import Fields.FieldOfFractions.Order ℤIntDom ℤOrderedRing
 ℚ : Set
 ℚ = fieldOfFractionsSet
 
+ℚSetoid : Setoid ℚ
+ℚSetoid = fieldOfFractionsSetoid
+
 _+Q_ : ℚ → ℚ → ℚ
 a +Q b = fieldOfFractionsPlus a b
 
@@ -43,6 +45,9 @@ a *Q b = fieldOfFractionsTimes a b
 injectionQ : ℤ → ℚ
 injectionQ z = z ,, (nonneg 1 , λ ())
 
+injectionQInjective : Injection injectionQ
+injectionQInjective pr = equalityLeft pr
+
 ℚField : Field ℚRing
 ℚField = fieldOfFractions
 
diff --git a/Numbers/Reals/Definition.agda b/Numbers/Reals/Definition.agda
index 8633d0c..a4f479c 100644
--- a/Numbers/Reals/Definition.agda
+++ b/Numbers/Reals/Definition.agda
@@ -1,9 +1,11 @@
-{-# OPTIONS --warning=error --safe --guardedness #-}
+{-# OPTIONS --warning=error --safe --guardedness --without-K #-}
 
 open import Setoids.Orders
+open import Setoids.Setoids
 open import LogicalFormulae
 open import Rings.Definition
 open import Numbers.Rationals.Definition
+open import Functions
 
 module Numbers.Reals.Definition where
 
@@ -23,12 +25,20 @@ _+R_ = _+C_
 _*R_ : ℝ → ℝ → ℝ
 _*R_ = _*C_
 
+ℝSetoid = cauchyCompletionSetoid
+
+_=R_ : ℝ → ℝ → Set
+a =R b = Setoid._∼_ cauchyCompletionSetoid a b
+
 ℝRing : Ring cauchyCompletionSetoid _+R_ _*R_
 ℝRing = CRing
 
 injectionR : ℚ → ℝ
 injectionR = injection
 
+injectionRInjective : Injection injectionR
+injectionRInjective = CInjection'
+
 0R : ℝ
 0R = injection 0Q
 
diff --git a/RedBlackTree.agda b/RedBlackTree.agda
deleted file mode 100644
index bbeb4fc..0000000
--- a/RedBlackTree.agda
+++ /dev/null
@@ -1,42 +0,0 @@
-{-# OPTIONS --warning=error --safe #-}
-
-open import LogicalFormulae
-open import Functions
-open import Maybe
-open import Orders
-
-open import Numbers.Naturals.Naturals
-
-module RedBlackTree where
-  record BinaryTreeNode {a : _} (carrier : Set a) (order : TotalOrder carrier) (minValue : Maybe carrier) (maxValue : Maybe carrier) : Set a
-
-  valueExtractor : {a : _} {carrier : Set a} {order : TotalOrder carrier} {minValue : Maybe carrier} {maxValue : Maybe carrier} → BinaryTreeNode {a} carrier order minValue maxValue → carrier
-
-  record BinaryTreeNode {a} carrier order minValue maxValue where
-    inductive
-    field
-      value : carrier
-      min<=val : TotalOrder._<_ order minValue maxValue
-      leftChild : Maybe (Sg (BinaryTreeNode {a} carrier order minValue (yes value)) (λ i → TotalOrder._<_ order (valueExtractor {a} {carrier} {order} i) value))
-      rightChild : Maybe (Sg (BinaryTreeNode {a} carrier order (yes value) maxValue) (λ i → TotalOrder._<_ order value (valueExtractor i)))
-
-  valueExtractor t = BinaryTreeNode.value t
-
-  lookup : {a : _} {carrier : Set a} {order : TotalOrder carrier} {minValue : Maybe carrier} {maxValue : Maybe carrier} → (t : BinaryTreeNode carrier order minValue maxValue) → (k : carrier) → Maybe carrier
-  lookup {a} {carrier} {order} record { value = value ; leftChild = leftChild ; rightChild = rightChild } k with TotalOrder.totality order k value
-  lookup {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = rightChild } k | inl (inl k<val) = no
-  lookup {a} {carrier} {order} record { value = value ; leftChild = (yes (tree , _)) ; rightChild = rightChild } k | inl (inl k<val) = lookup tree k
-  lookup {a} {carrier} {order} record { value = value ; leftChild = leftChild ; rightChild = no } k | inl (inr val<k) = no
-  lookup {a} {carrier} {order} record { value = value ; leftChild = leftChild ; rightChild = (yes (tree , _)) } k | inl (inr val<k) = lookup tree k
-  lookup {a} {carrier} {order} record { value = value ; leftChild = leftChild ; rightChild = rightChild } k | inr k=val = yes value
-
-  addTree : {a : _} {carrier : Set a} {order : TotalOrder carrier} {minValue : Maybe carrier} {maxValue : Maybe carrier} → (t : BinaryTreeNode carrier order minValue maxValue) → (k : carrier) → BinaryTreeNode carrier order (yes (defaultValue k minValue)) (yes (defaultValue k maxValue))
-  addTree {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = no } k with TotalOrder.totality order k value
-  addTree {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = no } k | inl (inr val<k) = record { value = value ; leftChild = no ; rightChild = yes (record { value = k ; leftChild = no ; rightChild = no} , val<k)}
-  addTree {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = no } k | inl (inl k<val) = record { value = value ; leftChild = yes (record { value = k ; leftChild = no ; rightChild = no} , k<val) ; rightChild = no }
-  addTree {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = no } k | inr k=val = record { value = k ; leftChild = no ; rightChild = no }
-  addTree {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = (yes x) } k with TotalOrder.totality order k value
-  addTree {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = (yes child) } k | inl (inl k<val) = {!!}
-  addTree {a} {carrier} {order} record { value = value ; leftChild = no ; rightChild = (yes child) } k | inl (inr val<k) = {!!}
-  addTree {a} {carrier} {order} {minValue} {maxValue} record { value = value ; leftChild = no ; rightChild = (yes (child , pr)) } k | inr k=val = record { value = k ; leftChild = no ; rightChild = yes (typeCast child (applyEquality (λ i → BinaryTreeNode carrier order {!!} {!!}) {!!}) , {!!}) }
-  addTree {a} {carrier} {order} record { value = value ; leftChild = (yes x) ; rightChild = rightChild } k = {!!}
diff --git a/Rings/Orders/Total/Lemmas.agda b/Rings/Orders/Total/Lemmas.agda
index f338bbd..cece1b6 100644
--- a/Rings/Orders/Total/Lemmas.agda
+++ b/Rings/Orders/Total/Lemmas.agda
@@ -293,6 +293,9 @@ abstract
   halvePositive a 0<2a | inl (inr a<0) = exFalso (irreflexive {a + a} (SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.reflexive eq) identRight (ringAddInequalities a<0 a<0)) 0<2a))
   halvePositive a 0<2a | inr x = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq x) (Equivalence.symmetric eq x)) identRight) 0<2a))
 
+  halvePositive' : {a b : A} → (a + a) ∼ b → 0R < b → 0R < a
+  halvePositive' {a} {b} pr 0<b = halvePositive a (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq pr) 0<b)
+
   0<1 : (0R ∼ 1R → False) → 0R < 1R
   0<1 0!=1 with totality 0R 1R
   0<1 0!=1 | inl (inl x) = x
diff --git a/Sequences.agda b/Sequences.agda
index 9a81893..9f96bcb 100644
--- a/Sequences.agda
+++ b/Sequences.agda
@@ -13,6 +13,9 @@ record Sequence {a : _} (A : Set a) : Set a where
     head : A
     tail : Sequence A
 
+headInjective : {a : _} {A : Set a} {s1 s2 : Sequence A} → s1 ≡ s2 → Sequence.head s1 ≡ Sequence.head s2
+headInjective {s1 = s1} {.s1} refl = refl
+
 constSequence : {a : _} {A : Set a} (k : A) → Sequence A
 Sequence.head (constSequence k) = k
 Sequence.tail (constSequence k) = constSequence k
diff --git a/Sets/Cardinality/Infinite/Examples.agda b/Sets/Cardinality/Infinite/Examples.agda
index 13a31a7..8ad048f 100644
--- a/Sets/Cardinality/Infinite/Examples.agda
+++ b/Sets/Cardinality/Infinite/Examples.agda
@@ -1,4 +1,4 @@
-{-# OPTIONS --safe --warning=error --without-K #-}
+{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
 
 open import Functions
 open import LogicalFormulae
@@ -8,6 +8,11 @@ open import Sets.FinSet.Definition
 open import Sets.FinSet.Lemmas
 open import Sets.Cardinality.Infinite.Definition
 open import Sets.Cardinality.Finite.Lemmas
+open import Numbers.Reals.Definition
+open import Numbers.Rationals.Definition
+open import Numbers.Integers.Definition
+open import Sets.Cardinality.Infinite.Lemmas
+open import Setoids.Setoids
 
 module Sets.Cardinality.Infinite.Examples where
 
@@ -28,3 +33,7 @@ module Sets.Cardinality.Infinite.Examples where
     bad a = (inv (toNat a))
     badInj : Injection bad
     badInj = injComp nextInj invInj
+
+ℝIsInfinite : DedekindInfiniteSet ℝ
+DedekindInfiniteSet.inj ℝIsInfinite n = injectionR (injectionQ (nonneg n))
+DedekindInfiniteSet.isInjection ℝIsInfinite {x} {y} pr = nonnegInjective (injectionQInjective (injectionRInjective pr))