From 660d7aa27c6dd5c90118a2fea19525f9c5622879 Mon Sep 17 00:00:00 2001
From: Patrick Stevens <smaug451@gmail.com>
Date: Fri, 22 Nov 2019 19:52:57 +0000
Subject: [PATCH] Lots of rings (#82)

---
 Everything/Safe.agda                          |   13 +-
 Fields/FieldOfFractions/Addition.agda         |    5 +-
 Fields/FieldOfFractions/Field.agda            |    2 +-
 Fields/FieldOfFractions/Group.agda            |    2 +-
 Fields/FieldOfFractions/Lemmas.agda           |    2 +-
 Fields/FieldOfFractions/Multiplication.agda   |    5 +-
 Fields/FieldOfFractions/Order.agda            | 1362 ++++++++---------
 Fields/FieldOfFractions/Ring.agda             |    2 +-
 Fields/FieldOfFractions/Setoid.agda           |   11 +-
 Fields/Fields.agda                            |    4 +-
 Fields/Orders/Lemmas.agda                     |   66 +-
 Fields/Orders/Partial/Definition.agda         |    1 -
 Fields/Orders/Total/Definition.agda           |    1 -
 Groups/Cosets.agda                            |    5 +
 Groups/FirstIsomorphismTheorem.agda           |   16 +-
 Groups/Groups.agda                            |   44 -
 Groups/QuotientGroup/Definition.agda          |   54 +-
 Groups/QuotientGroup/Lemmas.agda              |   17 +-
 Groups/Subgroups/Definition.agda              |   18 +-
 Groups/SymmetricGroups/Lemmas.agda            |    1 +
 Numbers/Integers/Multiplication.agda          |    1 -
 .../RingStructure/IntegralDomain.agda         |   12 +-
 Numbers/Integers/RingStructure/Ring.agda      |    1 -
 Rings/Cosets.agda                             |    5 +
 Rings/EuclideanDomains/Definition.agda        |   37 +
 Rings/EuclideanDomains/Examples.agda          |   39 +
 Rings/Homomorphisms/Definition.agda           |    1 -
 Rings/Homomorphisms/Image.agda                |   29 +
 Rings/Homomorphisms/Kernel.agda               |   21 +
 Rings/Ideals/Definition.agda                  |   31 -
 Rings/Ideals/FirstIsomorphismTheorem.agda     |   34 +
 Rings/Ideals/Lemmas.agda                      |   58 +
 Rings/Ideals/Prime/Definition.agda            |   22 +
 Rings/Ideals/Principal/Definition.agda        |   24 +
 Rings/IntegralDomains.agda                    |   37 -
 Rings/IntegralDomains/Definition.agda         |   41 +
 Rings/IntegralDomains/Examples.agda           |   28 +
 Rings/Isomorphisms/Definition.agda            |    7 +-
 Rings/PrincipalIdealDomain.agda               |   24 +
 Rings/Subrings/Definition.agda                |   44 +
 40 files changed, 1246 insertions(+), 881 deletions(-)
 create mode 100644 Rings/EuclideanDomains/Definition.agda
 create mode 100644 Rings/EuclideanDomains/Examples.agda
 create mode 100644 Rings/Homomorphisms/Image.agda
 create mode 100644 Rings/Homomorphisms/Kernel.agda
 create mode 100644 Rings/Ideals/FirstIsomorphismTheorem.agda
 create mode 100644 Rings/Ideals/Lemmas.agda
 create mode 100644 Rings/Ideals/Prime/Definition.agda
 create mode 100644 Rings/Ideals/Principal/Definition.agda
 delete mode 100644 Rings/IntegralDomains.agda
 create mode 100644 Rings/IntegralDomains/Definition.agda
 create mode 100644 Rings/IntegralDomains/Examples.agda
 create mode 100644 Rings/PrincipalIdealDomain.agda
 create mode 100644 Rings/Subrings/Definition.agda

diff --git a/Everything/Safe.agda b/Everything/Safe.agda
index 027c39d..44d80c4 100644
--- a/Everything/Safe.agda
+++ b/Everything/Safe.agda
@@ -47,7 +47,7 @@ open import Rings.Lemmas
 open import Rings.Orders.Partial.Definition
 open import Rings.Orders.Total.Lemmas
 open import Rings.Orders.Partial.Lemmas
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Rings.DirectSum
 open import Rings.Polynomial.Ring
 open import Rings.Polynomial.Evaluation
@@ -55,6 +55,17 @@ open import Rings.Ideals.Definition
 open import Rings.Isomorphisms.Definition
 open import Rings.Quotients.Definition
 open import Rings.Cosets
+open import Rings.EuclideanDomains.Definition
+open import Rings.EuclideanDomains.Examples
+open import Rings.Homomorphisms.Image
+open import Rings.Homomorphisms.Kernel
+open import Rings.Ideals.FirstIsomorphismTheorem
+open import Rings.Ideals.Lemmas
+open import Rings.Ideals.Prime.Definition
+open import Rings.Ideals.Principal.Definition
+open import Rings.IntegralDomains.Examples
+open import Rings.PrincipalIdealDomain
+open import Rings.Subrings.Definition
 
 open import Setoids.Setoids
 open import Setoids.DirectSum
diff --git a/Fields/FieldOfFractions/Addition.agda b/Fields/FieldOfFractions/Addition.agda
index 3f91267..04a0b5e 100644
--- a/Fields/FieldOfFractions/Addition.agda
+++ b/Fields/FieldOfFractions/Addition.agda
@@ -6,7 +6,7 @@ open import Groups.Definition
 open import Groups.Lemmas
 open import Rings.Definition
 open import Rings.Lemmas
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
@@ -25,8 +25,7 @@ fieldOfFractionsPlus (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (((a * d) + (b * c))
     open Ring R
     ans : ((b * d) ∼ Ring.0R R) → False
     ans pr with IntegralDomain.intDom I pr
-    ans pr | inl x = b!=0 x
-    ans pr | inr x = d!=0 x
+    ans pr | f = exFalso (d!=0 (f b!=0))
 
 plusWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus a b) (fieldOfFractionsPlus c d)
 plusWellDefined {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} {g ,, (h , h!=0)} af=be ch=dg = need
diff --git a/Fields/FieldOfFractions/Field.agda b/Fields/FieldOfFractions/Field.agda
index b5b5ec6..441be2a 100644
--- a/Fields/FieldOfFractions/Field.agda
+++ b/Fields/FieldOfFractions/Field.agda
@@ -6,7 +6,7 @@ open import Groups.Definition
 open import Groups.Lemmas
 open import Rings.Definition
 open import Rings.Lemmas
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
diff --git a/Fields/FieldOfFractions/Group.agda b/Fields/FieldOfFractions/Group.agda
index c73d7d7..1fee587 100644
--- a/Fields/FieldOfFractions/Group.agda
+++ b/Fields/FieldOfFractions/Group.agda
@@ -6,7 +6,7 @@ open import Groups.Definition
 open import Groups.Lemmas
 open import Rings.Definition
 open import Rings.Lemmas
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
diff --git a/Fields/FieldOfFractions/Lemmas.agda b/Fields/FieldOfFractions/Lemmas.agda
index 979d4e0..f352b4c 100644
--- a/Fields/FieldOfFractions/Lemmas.agda
+++ b/Fields/FieldOfFractions/Lemmas.agda
@@ -8,7 +8,7 @@ open import Groups.Lemmas
 open import Rings.Definition
 open import Rings.Lemmas
 open import Rings.Homomorphisms.Definition
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
diff --git a/Fields/FieldOfFractions/Multiplication.agda b/Fields/FieldOfFractions/Multiplication.agda
index 14911d5..1c075df 100644
--- a/Fields/FieldOfFractions/Multiplication.agda
+++ b/Fields/FieldOfFractions/Multiplication.agda
@@ -6,7 +6,7 @@ open import Groups.Definition
 open import Groups.Lemmas
 open import Rings.Definition
 open import Rings.Lemmas
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
@@ -25,8 +25,7 @@ fieldOfFractionsTimes (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (a * c) ,, ((b * d)
     open Ring R
     ans : ((b * d) ∼ Ring.0R R) → False
     ans pr with IntegralDomain.intDom I pr
-    ans pr | inl x = b!=0 x
-    ans pr | inr x = d!=0 x
+    ans pr | f = exFalso (d!=0 (f b!=0))
 
 fieldOfFractionsTimesWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → (Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsTimes a b) (fieldOfFractionsTimes c d))
 fieldOfFractionsTimesWellDefined {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need
diff --git a/Fields/FieldOfFractions/Order.agda b/Fields/FieldOfFractions/Order.agda
index 22b37eb..c1c9cad 100644
--- a/Fields/FieldOfFractions/Order.agda
+++ b/Fields/FieldOfFractions/Order.agda
@@ -9,7 +9,7 @@ open import Rings.Orders.Partial.Definition
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Total.Lemmas
 open import Rings.Lemmas
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
@@ -20,698 +20,694 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 
 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.Setoid I
+open import Fields.FieldOfFractions.Ring I
 
-  open SetoidTotalOrder (TotallyOrderedRing.total order)
-  open import Rings.Orders.Partial.Lemmas
-  open PartiallyOrderedRing pRing
+open SetoidTotalOrder (TotallyOrderedRing.total order)
+open import Rings.Orders.Partial.Lemmas
+open PartiallyOrderedRing pRing
 
-  fieldOfFractionsComparison : Rel fieldOfFractionsSet
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) with totality (Ring.0R R) denomA
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
-    where
-      open Equivalence (Setoid.eq S)
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) with totality (Ring.0R R) denomB
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
-    where
-      open Equivalence (Setoid.eq S)
-  fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
-    where
-      open Equivalence (Setoid.eq S)
+fieldOfFractionsComparison : Rel fieldOfFractionsSet
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) with totality (Ring.0R R) denomA
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
+  where
+    open Equivalence (Setoid.eq S)
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) with totality (Ring.0R R) denomB
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
+  where
+    open Equivalence (Setoid.eq S)
+fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
+  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)
+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)
 
-  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))
+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)
+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)
 
-  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))
+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))
 
-  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))
+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))
 
-  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} → (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} → (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) → 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) → (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) → (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)
+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)
 
-  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 | inl x = exFalso (denomB!=0 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 | inr x = exFalso (denomC!=0 x)
-  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)
-  ... | inl x = exFalso (denomB!=0 x)
-  ... | inr x = exFalso (denomC!=0 x)
-  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 | inl x = exFalso (denomA!=0 x)
-  PartiallyOrderedRing.orderRespectsAddition (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inr 0=dAdC | inr x = exFalso (denomC!=0 x)
-  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
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) with totality (Ring.0R R) denomA
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inl 0<dA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-      0<nA : 0R < numA
-      0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
-      0<nB : 0R < numB
-      0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
-      0<nAnB : 0R < (numA * numB)
-      0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive pRing 0<nB 0<nA)
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inr dA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dA<0 0<dB))))
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) with totality (Ring.0R R) denomA
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inl 0<dA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dB<0 0<dA))))
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inr dA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-      nB<0 : numB < 0R
-      nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
-      nA<0 : numA < 0R
-      nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
-      0<nAnB : 0R < (numA * numB)
-      0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative pRing nA<0 nB<0)
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with totality (Ring.0R R) denomB
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso f
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-      f : False
-      f with PartiallyOrderedRing.orderRespectsMultiplication pRing 0<denomA 0<denomB
-      ... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder bl dAdB<0)
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inr denomA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) ans
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-      0<nB : 0R < numB
-      0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
-      nA<0 : numA < 0R
-      nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
-      ans : (numA * numB) < 0R
-      ans = SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nA<0 0<nB)
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inl 0<denomA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) nAnB<0
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-      nB<0 : numB < 0R
-      nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
-      0<nA : 0R < numA
-      0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
-      nAnB<0 : (numA * numB) < 0R
-      nAnB<0 = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nB<0 0<nA)
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso f
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-      h : 0R < (denomA * denomB)
-      h = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing denomB<0 denomA<0)
-      f : False
-      f = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dAdB<0 h)
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inr 0=dAdB with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdB)
-  ... | inl x = exFalso (denomA!=0 x)
-  ... | inr x = exFalso (denomB!=0 x)
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 1<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) identIsIdent (ringCanMultiplyByNegative pRing 1<0 1<0))))
-    where
-      open Setoid S
-      open Equivalence (Setoid.eq S)
-      open Ring R
-  PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inr x = exFalso (IntegralDomain.nontrivial I (Equivalence.symmetric (Setoid.eq S) x))
+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.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
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) with totality (Ring.0R R) denomA
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inl 0<dA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+    0<nA : 0R < numA
+    0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
+    0<nB : 0R < numB
+    0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
+    0<nAnB : 0R < (numA * numB)
+    0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive pRing 0<nB 0<nA)
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inr dA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dA<0 0<dB))))
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) with totality (Ring.0R R) denomA
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inl 0<dA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dB<0 0<dA))))
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inr dA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+    nB<0 : numB < 0R
+    nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
+    nA<0 : numA < 0R
+    nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
+    0<nAnB : 0R < (numA * numB)
+    0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative pRing nA<0 nB<0)
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with totality (Ring.0R R) denomB
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso f
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+    f : False
+    f with PartiallyOrderedRing.orderRespectsMultiplication pRing 0<denomA 0<denomB
+    ... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder bl dAdB<0)
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inr denomA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) ans
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+    0<nB : 0R < numB
+    0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
+    nA<0 : numA < 0R
+    nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
+    ans : (numA * numB) < 0R
+    ans = SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nA<0 0<nB)
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inl 0<denomA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) nAnB<0
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+    nB<0 : numB < 0R
+    nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
+    0<nA : 0R < numA
+    0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
+    nAnB<0 : (numA * numB) < 0R
+    nAnB<0 = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nB<0 0<nA)
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso f
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+    h : 0R < (denomA * denomB)
+    h = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing denomB<0 denomA<0)
+    f : False
+    f = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dAdB<0 h)
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inr 0=dAdB with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdB)
+... | f = exFalso (denomB!=0 (f denomA!=0))
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 1<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) identIsIdent (ringCanMultiplyByNegative pRing 1<0 1<0))))
+  where
+    open Setoid S
+    open Equivalence (Setoid.eq S)
+    open Ring R
+PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inr x = exFalso (IntegralDomain.nontrivial I (Equivalence.symmetric (Setoid.eq S) x))
 
-  fieldOfFractionsOrderedRing : TotallyOrderedRing fieldOfFractionsPOrderedRing
-  TotallyOrderedRing.total fieldOfFractionsOrderedRing = fieldOfFractionsTotalOrder
+fieldOfFractionsOrderedRing : TotallyOrderedRing fieldOfFractionsPOrderedRing
+TotallyOrderedRing.total fieldOfFractionsOrderedRing = fieldOfFractionsTotalOrder
diff --git a/Fields/FieldOfFractions/Ring.agda b/Fields/FieldOfFractions/Ring.agda
index 1399959..a77fb00 100644
--- a/Fields/FieldOfFractions/Ring.agda
+++ b/Fields/FieldOfFractions/Ring.agda
@@ -6,7 +6,7 @@ open import Groups.Definition
 open import Groups.Lemmas
 open import Rings.Definition
 open import Rings.Lemmas
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
diff --git a/Fields/FieldOfFractions/Setoid.agda b/Fields/FieldOfFractions/Setoid.agda
index 54da07b..c124055 100644
--- a/Fields/FieldOfFractions/Setoid.agda
+++ b/Fields/FieldOfFractions/Setoid.agda
@@ -6,16 +6,17 @@ open import Groups.Definition
 open import Groups.Lemmas
 open import Rings.Definition
 open import Rings.Lemmas
-open import Rings.IntegralDomains
 open import Fields.Fields
 open import Functions
 open import Setoids.Setoids
 open import Sets.EquivalenceRelations
+open import Rings.IntegralDomains.Definition
 
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 
 module Fields.FieldOfFractions.Setoid {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
 
+
 fieldOfFractionsSet : Set (a ⊔ b)
 fieldOfFractionsSet = (A && (Sg A (λ a → (Setoid._∼_ S a (Ring.0R R) → False))))
 
@@ -36,9 +37,7 @@ Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {a ,, (b , b!=0)} {c ,
     p2 = transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive p (transitive (symmetric *Associative) (*WellDefined reflexive cf=de)))
     p3 : (a * f) * d ∼ (b * e) * d
     p3 = transitive p2 (transitive (*WellDefined reflexive *Commutative) *Associative)
-    p4 : (d ∼ 0R) || ((a * f) ∼ (b * e))
-    p4 = cancelIntDom I (transitive *Commutative (transitive p3 *Commutative))
+    p4 : ((d ∼ 0R) → False) → ((a * f) ∼ (b * e))
+    p4 = cancelIntDom R I (transitive *Commutative (transitive p3 *Commutative))
     p5 : (a * f) ∼ (b * e)
-    p5 with p4
-    p5 | inl d=0 = exFalso (d!=0 d=0)
-    p5 | inr x = x
+    p5 = p4 d!=0
diff --git a/Fields/Fields.agda b/Fields/Fields.agda
index e1d3b15..980fd1d 100644
--- a/Fields/Fields.agda
+++ b/Fields/Fields.agda
@@ -8,7 +8,7 @@ open import Rings.Lemmas
 open import Setoids.Setoids
 open import Setoids.Orders
 open import Orders
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Functions
 open import Sets.EquivalenceRelations
 
@@ -23,6 +23,8 @@ record Field {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A}
   field
     allInvertible : (a : A) → ((a ∼ Group.0G (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
     nontrivial : (0R ∼ 1R) → False
+  0F : A
+  0F = Ring.0R R
 
 record Field' {m n : _} : Set (lsuc m ⊔ lsuc n) where
   field
diff --git a/Fields/Orders/Lemmas.agda b/Fields/Orders/Lemmas.agda
index 2a514f2..20ec5a5 100644
--- a/Fields/Orders/Lemmas.agda
+++ b/Fields/Orders/Lemmas.agda
@@ -11,7 +11,7 @@ open import Rings.Lemmas
 open import Setoids.Setoids
 open import Setoids.Orders
 open import Orders
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 open import Functions
 open import Sets.EquivalenceRelations
 open import Fields.Fields
@@ -84,35 +84,39 @@ abstract
       r : (1/2 * ((1R + 1R) * x)) ∼ (1/2' * ((1R + 1R) * x))
       r = Equivalence.transitive eq *Associative (Equivalence.transitive eq q (Equivalence.symmetric eq *Associative))
 
+  private
+    orderedFieldIntDom : {a b : A} → (a * b ∼ 0R) → (a ∼ 0R) || (b ∼ 0R)
+    orderedFieldIntDom {a} {b} ab=0 with totality 0R a
+    ... | inl (inl x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
+      where
+        open Equivalence eq
+        a!=0 : (a ∼ Group.0G additiveGroup) → False
+        a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (symmetric pr) reflexive x)
+        invA : A
+        invA = underlying (Field.allInvertible F a a!=0)
+        q : 1R ∼ (invA * a)
+        q with Field.allInvertible F a a!=0
+        ... | invA , pr = symmetric pr
+        p : invA * (a * b) ∼ invA * Group.0G additiveGroup
+        p = *WellDefined reflexive ab=0
+        p' : (invA * a) * b ∼ Group.0G additiveGroup
+        p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
+    orderedFieldIntDom {a} {b} ab=0 | inl (inr x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
+      where
+        open Equivalence eq
+        a!=0 : (a ∼ Group.0G additiveGroup) → False
+        a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (symmetric pr) x)
+        invA : A
+        invA = underlying (Field.allInvertible F a a!=0)
+        q : 1R ∼ (invA * a)
+        q with Field.allInvertible F a a!=0
+        ... | invA , pr = symmetric pr
+        p : invA * (a * b) ∼ invA * Group.0G additiveGroup
+        p = *WellDefined reflexive ab=0
+        p' : (invA * a) * b ∼ Group.0G additiveGroup
+        p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
+    orderedFieldIntDom {a} {b} ab=0 | inr x = inl (Equivalence.symmetric (Setoid.eq S) x)
+
   orderedFieldIsIntDom :  IntegralDomain R
-  IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 with totality (Ring.0R R) a
-  IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 | inl (inl x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
-    where
-      open Equivalence eq
-      a!=0 : (a ∼ Group.0G additiveGroup) → False
-      a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (symmetric pr) reflexive x)
-      invA : A
-      invA = underlying (Field.allInvertible F a a!=0)
-      q : 1R ∼ (invA * a)
-      q with Field.allInvertible F a a!=0
-      ... | invA , pr = symmetric pr
-      p : invA * (a * b) ∼ invA * Group.0G additiveGroup
-      p = *WellDefined reflexive ab=0
-      p' : (invA * a) * b ∼ Group.0G additiveGroup
-      p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
-  IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 | inl (inr x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
-    where
-      open Equivalence eq
-      a!=0 : (a ∼ Group.0G additiveGroup) → False
-      a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (symmetric pr) x)
-      invA : A
-      invA = underlying (Field.allInvertible F a a!=0)
-      q : 1R ∼ (invA * a)
-      q with Field.allInvertible F a a!=0
-      ... | invA , pr = symmetric pr
-      p : invA * (a * b) ∼ invA * Group.0G additiveGroup
-      p = *WellDefined reflexive ab=0
-      p' : (invA * a) * b ∼ Group.0G additiveGroup
-      p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
-  IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 | inr x = inl (Equivalence.symmetric (Setoid.eq S) x)
+  IntegralDomain.intDom orderedFieldIsIntDom = decidedIntDom R orderedFieldIntDom
   IntegralDomain.nontrivial orderedFieldIsIntDom pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)
diff --git a/Fields/Orders/Partial/Definition.agda b/Fields/Orders/Partial/Definition.agda
index 71c4f5e..3ea5c37 100644
--- a/Fields/Orders/Partial/Definition.agda
+++ b/Fields/Orders/Partial/Definition.agda
@@ -9,7 +9,6 @@ open import Rings.Lemmas
 open import Setoids.Setoids
 open import Setoids.Orders
 open import Orders
-open import Rings.IntegralDomains
 open import Functions
 open import Sets.EquivalenceRelations
 open import Fields.Fields
diff --git a/Fields/Orders/Total/Definition.agda b/Fields/Orders/Total/Definition.agda
index 8fb6e6f..bf6fd38 100644
--- a/Fields/Orders/Total/Definition.agda
+++ b/Fields/Orders/Total/Definition.agda
@@ -10,7 +10,6 @@ open import Rings.Lemmas
 open import Setoids.Setoids
 open import Setoids.Orders
 open import Orders
-open import Rings.IntegralDomains
 open import Functions
 open import Sets.EquivalenceRelations
 open import Fields.Fields
diff --git a/Groups/Cosets.agda b/Groups/Cosets.agda
index 40218e3..65718ca 100644
--- a/Groups/Cosets.agda
+++ b/Groups/Cosets.agda
@@ -7,6 +7,7 @@ open import Setoids.Subset
 open import Functions
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import Groups.Definition
+open import Groups.Homomorphisms.Definition
 open import Groups.Subgroups.Definition
 open import Groups.Subgroups.Normal.Definition
 
@@ -45,3 +46,7 @@ Group.identRight (cosetGroup norm) = isSubset (symmetric (transitive +Associativ
 Group.identLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive identLeft) invLeft)) containsIdentity
 Group.invLeft (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invLeft) invLeft)) containsIdentity
 Group.invRight (cosetGroup norm) = isSubset (symmetric (transitive (+WellDefined reflexive invRight) invLeft)) containsIdentity
+
+cosetGroupHom : (norm : normalSubgroup G subgrp) → GroupHom G (cosetGroup norm) id
+GroupHom.groupHom (cosetGroupHom norm) = isSubset (symmetric (transitive (+WellDefined invContravariant reflexive) (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive invLeft)) reflexive) (transitive (+WellDefined identRight reflexive) invLeft))))) (Subgroup.containsIdentity subgrp)
+GroupHom.wellDefined (cosetGroupHom norm) {x} {y} x=y = isSubset (symmetric (transitive (+WellDefined reflexive x=y) invLeft)) (Subgroup.containsIdentity subgrp)
diff --git a/Groups/FirstIsomorphismTheorem.agda b/Groups/FirstIsomorphismTheorem.agda
index 3721b75..a0cbb30 100644
--- a/Groups/FirstIsomorphismTheorem.agda
+++ b/Groups/FirstIsomorphismTheorem.agda
@@ -18,7 +18,6 @@ open import Groups.Subgroups.Normal.Definition
 open import Groups.Subgroups.Normal.Lemmas
 open import Groups.Lemmas
 open import Groups.Abelian.Definition
-open import Groups.QuotientGroup.Definition
 
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 
@@ -27,6 +26,7 @@ module Groups.FirstIsomorphismTheorem {a b c d : _} {A : Set a} {B : Set b} {S :
 open import Groups.Homomorphisms.Image fHom
 open import Groups.Homomorphisms.Kernel fHom
 open import Groups.Cosets G groupKernelIsSubgroup
+open import Groups.QuotientGroup.Definition G fHom
 open Setoid T
 open Equivalence eq
 open import Setoids.Subset T
@@ -39,7 +39,7 @@ groupFirstIsomorphismTheoremWellDefined {x} {y} pr = transitive (symmetric (invT
     u : Setoid._∼_ T (Group.inverse H (Group.inverse H (f y))) (Group.inverse H (Group.inverse H (f x)))
     u = inverseWellDefined H (transferToRight' H t)
 
-groupFirstIsomorphismTheorem : GroupsIsomorphic (cosetGroup groupKernelIsNormalSubgroup) (subgroupIsGroup H imageGroupSubset imageGroupSubgroup)
+groupFirstIsomorphismTheorem : GroupsIsomorphic (cosetGroup groupKernelIsNormalSubgroup) (subgroupIsGroup H imageGroupSubgroup)
 GroupsIsomorphic.isomorphism groupFirstIsomorphismTheorem x = f x , (x , reflexive)
 GroupHom.groupHom (GroupIso.groupHom (GroupsIsomorphic.proof groupFirstIsomorphismTheorem)) {x} {y} = GroupHom.groupHom fHom
 GroupHom.wellDefined (GroupIso.groupHom (GroupsIsomorphic.proof groupFirstIsomorphismTheorem)) {x} {y} = groupFirstIsomorphismTheoremWellDefined {x} {y}
@@ -47,3 +47,15 @@ SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic
 SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem))) fx=fy = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H reflexive fx=fy) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom))))
 SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem))) = groupFirstIsomorphismTheoremWellDefined
 SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem))) {b , (a , fa=b)} = a , fa=b
+
+groupFirstIsomorphismTheoremWellDefined' : {x y : A} → f (x +G (Group.inverse G y)) ∼ Group.0G H → Setoid._∼_ (subsetSetoid imageGroupSubset) (f x , (x , reflexive)) (f y , (y , reflexive))
+groupFirstIsomorphismTheoremWellDefined' {x} {y} pr = transferToRight H (transitive (symmetric (transitive (GroupHom.groupHom fHom) (Group.+WellDefined H reflexive (homRespectsInverse fHom)))) pr)
+
+groupFirstIsomorphismTheorem' : GroupsIsomorphic (quotientGroupByHom) (subgroupIsGroup H imageGroupSubgroup)
+GroupsIsomorphic.isomorphism groupFirstIsomorphismTheorem' a = f a , (a , reflexive)
+GroupHom.groupHom (GroupIso.groupHom (GroupsIsomorphic.proof groupFirstIsomorphismTheorem')) {x} {y} = GroupHom.groupHom fHom
+GroupHom.wellDefined (GroupIso.groupHom (GroupsIsomorphic.proof groupFirstIsomorphismTheorem')) {x} {y} = groupFirstIsomorphismTheoremWellDefined'
+SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem'))) {x} {y} = groupFirstIsomorphismTheoremWellDefined' {x} {y}
+SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem'))) {x} {y} fx=fy = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H reflexive (homRespectsInverse fHom)) (transferToRight'' H fx=fy))
+SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem'))) {x} {y} = groupFirstIsomorphismTheoremWellDefined'
+SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem'))) {b , (a , fa=b)} = a , fa=b
diff --git a/Groups/Groups.agda b/Groups/Groups.agda
index 105acea..4dc6171 100644
--- a/Groups/Groups.agda
+++ b/Groups/Groups.agda
@@ -43,50 +43,6 @@ fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetr
     open Setoid S
     open Equivalence eq
 
-quotientGroupSetoid : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → (Setoid {a} {d} A)
-quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H} {f} fHom = ansSetoid
-  where
-    open Setoid T
-    open Group H
-    open Equivalence eq
-    ansSetoid : Setoid A
-    Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
-    Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
-    Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
-      where
-        g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
-        g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdent H))
-        h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
-        h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
-        i : f (n ·A Group.inverse G m) ∼ 0G
-        i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
-    Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
-      where
-        g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
-        g = replaceGroupOp H reflexive reflexive prmn prno reflexive
-        h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
-        h = transitive g identLeft
-        i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
-        i = transitive (GroupHom.groupHom fHom) h
-        j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
-        j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
-        k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
-        k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
-
-quotientGroupLemma : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ T (underf x) (underf y) → Setoid._∼_ (quotientGroupSetoid G f) x y
-quotientGroupLemma {S = S} {T = T} G {H = H} fHom {x} {y} fx=fy = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (homRespectsInverse fHom)) (transferToRight'' H fx=fy))
-  where
-    open Group G
-    open Setoid T
-    open Equivalence eq
-
-quotientGroupLemma' : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ (quotientGroupSetoid G f) x y → Setoid._∼_ T (underf x) (underf y)
-quotientGroupLemma' {S = S} {T = T} G {H = H} f fx=fy = transferToRight H (transitive (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (symmetric (homRespectsInverse f))) (symmetric (GroupHom.groupHom f))) fx=fy)
-  where
-    open Group G
-    open Setoid T
-    open Equivalence eq
-
 {-
 quotientHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {f : A → B} → (fHom : GroupHom G H f) → A → A
 quotientHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom a = {!!}
diff --git a/Groups/QuotientGroup/Definition.agda b/Groups/QuotientGroup/Definition.agda
index 6235ca8..f2c8f26 100644
--- a/Groups/QuotientGroup/Definition.agda
+++ b/Groups/QuotientGroup/Definition.agda
@@ -7,20 +7,48 @@ open import Groups.Definition
 open import Groups.Groups
 open import Groups.FiniteGroups.Definition
 open import Groups.Homomorphisms.Definition
-open import Groups.Abelian.Definition
 open import Setoids.Setoids
-open import Rings.Definition
-open import Fields.FieldOfFractions.Setoid
 open import Sets.EquivalenceRelations
 open import Groups.Lemmas
 open import Groups.Homomorphisms.Lemmas
 open import Groups.Subgroups.Definition
 open import Groups.Subgroups.Normal.Definition
 
-module Groups.QuotientGroup.Definition where
+module Groups.QuotientGroup.Definition {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (fHom : GroupHom G H f) where
 
-quotientGroupByHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
-Group.+WellDefined (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
+quotientGroupSetoid : (Setoid {a} {d} A)
+quotientGroupSetoid = ansSetoid
+  where
+    open Setoid T
+    open Group H
+    open Equivalence eq
+    ansSetoid : Setoid A
+    Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
+    Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
+    Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
+      where
+        g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
+        g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdent H))
+        h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
+        h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
+        i : f (n ·A Group.inverse G m) ∼ 0G
+        i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
+    Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
+      where
+        g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
+        g = replaceGroupOp H reflexive reflexive prmn prno reflexive
+        h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
+        h = transitive g identLeft
+        i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
+        i = transitive (GroupHom.groupHom fHom) h
+        j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
+        j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
+        k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
+        k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
+
+
+quotientGroupByHom : Group (quotientGroupSetoid) _·A_
+Group.+WellDefined (quotientGroupByHom) {x} {y} {m} {n} x~m y~n = ans
   where
     open Setoid T
     open Equivalence (Setoid.eq T)
@@ -42,15 +70,15 @@ Group.+WellDefined (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ =
     p8 = x~m
     ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
     ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
-Group.0G (quotientGroupByHom G fHom) = Group.0G G
-Group.inverse (quotientGroupByHom G fHom) = Group.inverse G
-Group.+Associative (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
+Group.0G (quotientGroupByHom) = Group.0G G
+Group.inverse (quotientGroupByHom) = Group.inverse G
+Group.+Associative (quotientGroupByHom) {a} {b} {c} = ans
   where
     open Setoid T
     open Equivalence (Setoid.eq T)
     ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
     ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
-Group.identRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
+Group.identRight (quotientGroupByHom) {a} = ans
   where
     open Group G
     open Setoid T
@@ -58,7 +86,7 @@ Group.identRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {
     transitiveG = Equivalence.transitive (Setoid.eq S)
     ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
     ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
-Group.identLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
+Group.identLeft (quotientGroupByHom) {a} = ans
   where
     open Group G
     open Setoid T
@@ -66,14 +94,14 @@ Group.identLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {u
     transitiveG = Equivalence.transitive (Setoid.eq S)
     ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
     ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
-Group.invLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
+Group.invLeft (quotientGroupByHom) {x} = ans
   where
     open Group G
     open Setoid T
     open Equivalence eq
     ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
     ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdent G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
-Group.invRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
+Group.invRight (quotientGroupByHom) {x} = ans
   where
     open Group G
     open Setoid T
diff --git a/Groups/QuotientGroup/Lemmas.agda b/Groups/QuotientGroup/Lemmas.agda
index 6edc3a6..f155d5f 100644
--- a/Groups/QuotientGroup/Lemmas.agda
+++ b/Groups/QuotientGroup/Lemmas.agda
@@ -9,13 +9,28 @@ open import Groups.FiniteGroups.Definition
 open import Groups.Homomorphisms.Definition
 open import Groups.Abelian.Definition
 open import Setoids.Setoids
-open import Fields.FieldOfFractions.Setoid
 open import Sets.EquivalenceRelations
 open import Groups.Lemmas
 open import Groups.QuotientGroup.Definition
+open import Groups.Homomorphisms.Lemmas
 
 module Groups.QuotientGroup.Lemmas {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} (G : Group S _+A_) (H : Group T _+B_) {f : A → B} (fHom : GroupHom G H f) where
 
+quotientGroupLemma : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ T (underf x) (underf y) → Setoid._∼_ (quotientGroupSetoid G f) x y
+quotientGroupLemma {S = S} {T = T} G {H = H} fHom {x} {y} fx=fy = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (homRespectsInverse fHom)) (transferToRight'' H fx=fy))
+  where
+    open Group G
+    open Setoid T
+    open Equivalence eq
+
+quotientGroupLemma' : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ (quotientGroupSetoid G f) x y → Setoid._∼_ T (underf x) (underf y)
+quotientGroupLemma' {S = S} {T = T} G {H = H} f fx=fy = transferToRight H (transitive (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (symmetric (homRespectsInverse f))) (symmetric (GroupHom.groupHom f))) fx=fy)
+  where
+    open Group G
+    open Setoid T
+    open Equivalence eq
+
+
 projectionMapIsGroupHom : GroupHom G (quotientGroupByHom G fHom) id
 GroupHom.groupHom projectionMapIsGroupHom {x} {y} = quotientGroupLemma G fHom (Equivalence.reflexive (Setoid.eq T))
 GroupHom.wellDefined projectionMapIsGroupHom x=y = quotientGroupLemma G fHom (GroupHom.wellDefined fHom x=y)
diff --git a/Groups/Subgroups/Definition.agda b/Groups/Subgroups/Definition.agda
index aa276d8..b10e4db 100644
--- a/Groups/Subgroups/Definition.agda
+++ b/Groups/Subgroups/Definition.agda
@@ -24,12 +24,12 @@ record Subgroup {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
 subgroupOp : {c : _} {pred : A → Set c} → (s : Subgroup pred) → Sg A pred → Sg A pred → Sg A pred
 subgroupOp {pred = pred} record { closedUnderPlus = one } (a , prA) (b , prB) = (a + b) , one prA prB
 
-subgroupIsGroup : {c : _} {pred : A → Set c} → (subs : subset pred) → (s : Subgroup pred) → Group (subsetSetoid subs) (subgroupOp s)
-Group.+WellDefined (subgroupIsGroup _ s) {m , prM} {n , prN} {x , prX} {y , prY} m=x n=y = +WellDefined m=x n=y
-Group.0G (subgroupIsGroup _ record { containsIdentity = two }) = 0G , two
-Group.inverse (subgroupIsGroup _ record { closedUnderInverse = three }) (a , prA) = (inverse a) , three prA
-Group.+Associative (subgroupIsGroup _ s) {a , prA} {b , prB} {c , prC} = +Associative
-Group.identRight (subgroupIsGroup _ s) {a , prA} = identRight
-Group.identLeft (subgroupIsGroup _ s) {a , prA} = identLeft
-Group.invLeft (subgroupIsGroup _ s) {a , prA} = invLeft
-Group.invRight (subgroupIsGroup _ s) {a , prA} = invRight
+subgroupIsGroup : {c : _} {pred : A → Set c} → (s : Subgroup pred) → Group (subsetSetoid (Subgroup.isSubset s)) (subgroupOp s)
+Group.+WellDefined (subgroupIsGroup s) {m , prM} {n , prN} {x , prX} {y , prY} m=x n=y = +WellDefined m=x n=y
+Group.0G (subgroupIsGroup record { containsIdentity = two }) = 0G , two
+Group.inverse (subgroupIsGroup record { closedUnderInverse = three }) (a , prA) = (inverse a) , three prA
+Group.+Associative (subgroupIsGroup s) {a , prA} {b , prB} {c , prC} = +Associative
+Group.identRight (subgroupIsGroup s) {a , prA} = identRight
+Group.identLeft (subgroupIsGroup s) {a , prA} = identLeft
+Group.invLeft (subgroupIsGroup s) {a , prA} = invLeft
+Group.invRight (subgroupIsGroup s) {a , prA} = invRight
diff --git a/Groups/SymmetricGroups/Lemmas.agda b/Groups/SymmetricGroups/Lemmas.agda
index 061efd8..3f3d81d 100644
--- a/Groups/SymmetricGroups/Lemmas.agda
+++ b/Groups/SymmetricGroups/Lemmas.agda
@@ -11,6 +11,7 @@ open import Groups.Lemmas
 open import Groups.Groups
 open import Groups.Subgroups.Definition
 open import Groups.Homomorphisms.Definition
+open import Groups.QuotientGroup.Definition
 open import Groups.Homomorphisms.Lemmas
 open import Groups.Actions.Definition
 open import Sets.EquivalenceRelations
diff --git a/Numbers/Integers/Multiplication.agda b/Numbers/Integers/Multiplication.agda
index a835dbb..13b9a41 100644
--- a/Numbers/Integers/Multiplication.agda
+++ b/Numbers/Integers/Multiplication.agda
@@ -9,7 +9,6 @@ open import Semirings.Definition
 open import Groups.Definition
 open import Rings.Definition
 open import Setoids.Setoids
-open import Rings.IntegralDomains
 
 module Numbers.Integers.Multiplication where
 
diff --git a/Numbers/Integers/RingStructure/IntegralDomain.agda b/Numbers/Integers/RingStructure/IntegralDomain.agda
index e1637d0..42137f7 100644
--- a/Numbers/Integers/RingStructure/IntegralDomain.agda
+++ b/Numbers/Integers/RingStructure/IntegralDomain.agda
@@ -8,15 +8,15 @@ open import Semirings.Definition
 open import Groups.Definition
 open import Rings.Definition
 open import Setoids.Setoids
-open import Rings.IntegralDomains
+open import Rings.IntegralDomains.Definition
 
 module Numbers.Integers.RingStructure.IntegralDomain where
 
-intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0) || (b ≡ nonneg 0)
-intDom (nonneg zero) (nonneg b) pr = inl refl
-intDom (nonneg (succ a)) (nonneg zero) pr = inr refl
-intDom (nonneg zero) (negSucc b) pr = inl refl
-intDom (negSucc a) (nonneg zero) pr = inr refl
+intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0 → False) → (b ≡ nonneg 0)
+intDom (nonneg zero) (nonneg b) x a!=0 = exFalso (a!=0 x)
+intDom (nonneg (succ a)) (nonneg zero) a!=0 _ = refl
+intDom (nonneg zero) (negSucc b) pr a!=0 = exFalso (a!=0 pr)
+intDom (negSucc a) (nonneg zero) _ _ = refl
 
 ℤIntDom : IntegralDomain ℤRing
 IntegralDomain.intDom ℤIntDom {a} {b} = intDom a b
diff --git a/Numbers/Integers/RingStructure/Ring.agda b/Numbers/Integers/RingStructure/Ring.agda
index b490855..6a6807b 100644
--- a/Numbers/Integers/RingStructure/Ring.agda
+++ b/Numbers/Integers/RingStructure/Ring.agda
@@ -7,7 +7,6 @@ open import Semirings.Definition
 open import Groups.Definition
 open import Rings.Definition
 open import Setoids.Setoids
-open import Rings.IntegralDomains
 
 module Numbers.Integers.RingStructure.Ring where
 
diff --git a/Rings/Cosets.agda b/Rings/Cosets.agda
index 5f189f8..00df2ca 100644
--- a/Rings/Cosets.agda
+++ b/Rings/Cosets.agda
@@ -47,3 +47,8 @@ Ring.*Associative cosetRing = isSubset (transitive (symmetric invLeft) (+WellDef
 Ring.*Commutative cosetRing {a} {b} = isSubset (transitive (symmetric invLeft) (+WellDefined (inverseWellDefined additiveGroup *Commutative) reflexive)) containsIdentity
 Ring.*DistributesOver+ cosetRing = isSubset (symmetric (transitive (+WellDefined (inverseWellDefined additiveGroup (symmetric *DistributesOver+)) reflexive) invLeft)) containsIdentity
 Ring.identIsIdent cosetRing = isSubset (symmetric (transitive (Group.+WellDefined additiveGroup reflexive identIsIdent) invLeft)) containsIdentity
+
+cosetRingHom : RingHom R cosetRing id
+RingHom.preserves1 cosetRingHom = isSubset (symmetric invLeft) (Ideal.containsIdentity ideal)
+RingHom.ringHom cosetRingHom = isSubset (symmetric invLeft) (Ideal.containsIdentity ideal)
+RingHom.groupHom cosetRingHom = cosetGroupHom (abelianGroupSubgroupIsNormal (Ideal.isSubgroup ideal) abelianUnderlyingGroup)
diff --git a/Rings/EuclideanDomains/Definition.agda b/Rings/EuclideanDomains/Definition.agda
new file mode 100644
index 0000000..ac9e9ac
--- /dev/null
+++ b/Rings/EuclideanDomains/Definition.agda
@@ -0,0 +1,37 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Definition
+open import Numbers.Naturals.Order
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+open import Rings.IntegralDomains.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.EuclideanDomains.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
+
+open Setoid S
+open Ring R
+
+record DivisionAlgorithmResult {norm : {a : A} → ((a ∼ 0R) → False) → ℕ} {x y : A} (x!=0 : (x ∼ 0R) → False) (y!=0 : (y ∼ 0R) → False) : Set (a ⊔ b) where
+  field
+    quotient : A
+    rem : A
+    remSmall : (rem ∼ 0R) || Sg ((rem ∼ 0R) → False) (λ rem!=0 → (norm rem!=0) <N (norm y!=0))
+    divAlg : x ∼ ((quotient * y) + rem)
+
+record EuclideanDomain : Set (a ⊔ lsuc b) where
+  field
+    isIntegralDomain : IntegralDomain R
+    norm : {a : A} → ((a ∼ 0R) → False) → ℕ
+    normSize : {a b : A} → (a!=0 : (a ∼ 0R) → False) → (b!=0 : (b ∼ 0R) → False) → (c : A) → b ∼ (a * c) → (norm a!=0) ≤N (norm b!=0)
+    divisionAlg : {a b : A} → (a!=0 : (a ∼ 0R) → False) → (b!=0 : (b ∼ 0R) → False) → DivisionAlgorithmResult {norm} a!=0 b!=0
diff --git a/Rings/EuclideanDomains/Examples.agda b/Rings/EuclideanDomains/Examples.agda
new file mode 100644
index 0000000..80605e2
--- /dev/null
+++ b/Rings/EuclideanDomains/Examples.agda
@@ -0,0 +1,39 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Definition
+open import Numbers.Naturals.Order
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+open import Rings.IntegralDomains.Definition
+open import Rings.IntegralDomains.Examples
+open import Rings.EuclideanDomains.Definition
+open import Fields.Fields
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.EuclideanDomains.Examples where
+
+polynomialField : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → (Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → EuclideanDomain R
+EuclideanDomain.isIntegralDomain (polynomialField F 1!=0) = fieldIsIntDom F 1!=0
+EuclideanDomain.norm (polynomialField F _) a!=0 = zero
+EuclideanDomain.normSize (polynomialField F _) a!=0 b!=0 c b=ac = inr refl
+DivisionAlgorithmResult.quotient (EuclideanDomain.divisionAlg (polynomialField {_*_ = _*_} F _) {a = a} {b} a!=0 b!=0) with Field.allInvertible F b b!=0
+... | bInv , prB = a * bInv
+DivisionAlgorithmResult.rem (EuclideanDomain.divisionAlg (polynomialField F _) a!=0 b!=0) = Field.0F F
+DivisionAlgorithmResult.remSmall (EuclideanDomain.divisionAlg (polynomialField {S = S} F _) a!=0 b!=0) = inl (Equivalence.reflexive (Setoid.eq S))
+DivisionAlgorithmResult.divAlg (EuclideanDomain.divisionAlg (polynomialField {S = S} {R = R} F _) {a = a} {b = b} a!=0 b!=0) with Field.allInvertible F b b!=0
+... | bInv , prB = transitive (transitive (transitive (symmetric identIsIdent) (transitive *Commutative (*WellDefined reflexive (symmetric prB)))) *Associative) (symmetric identRight)
+  where
+    open Setoid S
+    open Equivalence eq
+    open Ring R
+    open Group additiveGroup
diff --git a/Rings/Homomorphisms/Definition.agda b/Rings/Homomorphisms/Definition.agda
index 4d991e8..5b77e67 100644
--- a/Rings/Homomorphisms/Definition.agda
+++ b/Rings/Homomorphisms/Definition.agda
@@ -13,7 +13,6 @@ open import Rings.Definition
 
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 
--- Following Part IB's course Groups, Rings, and Modules, we take rings to be commutative with one.
 module Rings.Homomorphisms.Definition where
 
 record RingHom {m n o p : _} {A : Set m} {B : Set n} {SA : Setoid {m} {o} A} {SB : Setoid {n} {p} B} {_+A_ : A → A → A} {_*A_ : A → A → A} (R : Ring SA _+A_ _*A_) {_+B_ : B → B → B} {_*B_ : B → B → B} (S : Ring SB _+B_ _*B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
diff --git a/Rings/Homomorphisms/Image.agda b/Rings/Homomorphisms/Image.agda
new file mode 100644
index 0000000..dedf3a8
--- /dev/null
+++ b/Rings/Homomorphisms/Image.agda
@@ -0,0 +1,29 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Homomorphisms.Image {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_+A_ _*A_ : A → A → A} {_+B_ _*B_ : B → B → B} {R1 : Ring S _+A_ _*A_} {R2 : Ring T _+B_ _*B_} {f : A → B} (hom : RingHom R1 R2 f) where
+
+open import Groups.Homomorphisms.Image (RingHom.groupHom hom)
+open import Rings.Subrings.Definition
+
+imageGroupSubring : Subring R2 imageGroupPred
+Subring.isSubgroup imageGroupSubring = imageGroupSubgroup
+Subring.containsOne imageGroupSubring = Ring.1R R1 , RingHom.preserves1 hom
+Subring.closedUnderProduct imageGroupSubring {x} {y} (a , fa=x) (b , fb=y) = (a *A b) , transitive ringHom (Ring.*WellDefined R2 fa=x fb=y)
+  where
+    open Setoid T
+    open Equivalence eq
+    open RingHom hom
diff --git a/Rings/Homomorphisms/Kernel.agda b/Rings/Homomorphisms/Kernel.agda
new file mode 100644
index 0000000..a68ba47
--- /dev/null
+++ b/Rings/Homomorphisms/Kernel.agda
@@ -0,0 +1,21 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Homomorphisms.Kernel {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_+1_ _*1_ : A → A → A} {_+2_ _*2_ : B → B → B} {R1 : Ring S _+1_ _*1_} {R2 : Ring T _+2_ _*2_} {f : A → B} (fHom : RingHom R1 R2 f) where
+
+ringKernel : Set (a ⊔ d)
+ringKernel = Sg A (λ a → Setoid._∼_ T (f a) (Ring.0R R2))
diff --git a/Rings/Ideals/Definition.agda b/Rings/Ideals/Definition.agda
index 439b058..e037428 100644
--- a/Rings/Ideals/Definition.agda
+++ b/Rings/Ideals/Definition.agda
@@ -19,9 +19,6 @@ module Rings.Ideals.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_
 
 open import Groups.Subgroups.Definition (Ring.additiveGroup R)
 
-ringKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → Set (a ⊔ d)
-ringKernel {T = T} R2 {f} fHom = Sg A (λ a → Setoid._∼_ T (f a) (Ring.0R R2))
-
 record Ideal {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
   field
     isSubgroup : Subgroup pred
@@ -30,31 +27,3 @@ record Ideal {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
   closedUnderInverse = Subgroup.closedUnderInverse isSubgroup
   containsIdentity = Subgroup.containsIdentity isSubgroup
   isSubset = Subgroup.isSubset isSubgroup
-
-idealPredForKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → A → Set d
-idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
-
-idealPredForKernelWellDefined : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → {x y : A} → (Setoid._∼_ S x y) → (idealPredForKernel R2 fHom x → idealPredForKernel R2 fHom y)
-idealPredForKernelWellDefined {T = T} R2 {f} fHom a x=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined (RingHom.groupHom fHom) (Equivalence.symmetric (Setoid.eq S) a)) x=0
-
-kernelIdealIsIdeal : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} {R2 : Ring T _+2_ _*2_} {f : A → C} (fHom : RingHom R R2 f) → Ideal (idealPredForKernel R2 fHom)
-Subgroup.isSubset (Ideal.isSubgroup (kernelIdealIsIdeal {R2 = R2} fHom)) = idealPredForKernelWellDefined R2 fHom
-Subgroup.closedUnderPlus (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} {y} fx=0 fy=0 = transitive (transitive (GroupHom.groupHom (RingHom.groupHom fHom)) (+WellDefined fx=0 fy=0)) identLeft
-  where
-    open Ring R2
-    open Group (Ring.additiveGroup R2)
-    open Setoid T
-    open Equivalence eq
-Subgroup.containsIdentity (Ideal.isSubgroup (kernelIdealIsIdeal fHom)) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
-Subgroup.closedUnderInverse (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} fx=0 = zeroImpliesInverseZero (RingHom.groupHom fHom) fx=0
-  where
-    open Ring R2
-    open Group (Ring.additiveGroup R2)
-    open Setoid T
-    open Equivalence eq
-Ideal.accumulatesTimes (kernelIdealIsIdeal {T = T} {R2 = R2} {f = f} fHom) {x} {y} fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2 {f y})))
-  where
-    open Setoid T
-    open Equivalence eq
-
--- TODO : define the quotient by an ideal; note that the result is a ring
diff --git a/Rings/Ideals/FirstIsomorphismTheorem.agda b/Rings/Ideals/FirstIsomorphismTheorem.agda
new file mode 100644
index 0000000..19e1a22
--- /dev/null
+++ b/Rings/Ideals/FirstIsomorphismTheorem.agda
@@ -0,0 +1,34 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+open import Rings.Subrings.Definition
+open import Rings.Isomorphisms.Definition
+open import Groups.Isomorphisms.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Ideals.FirstIsomorphismTheorem {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_+A_ _*A_ : A → A → A} {_+B_ _*B_ : B → B → B} {R1 : Ring S _+A_ _*A_} {R2 : Ring T _+B_ _*B_} {f : A → B} (hom : RingHom R1 R2 f) where
+
+open import Rings.Quotients.Definition R1 R2 hom
+open import Rings.Homomorphisms.Image hom
+open Setoid T
+open Equivalence eq
+open import Groups.FirstIsomorphismTheorem (RingHom.groupHom hom)
+
+ringFirstIsomorphismTheorem' : RingsIsomorphic quotientByRingHom (subringIsRing R2 imageGroupSubring)
+RingsIsomorphic.f ringFirstIsomorphismTheorem' a = f a , (a , reflexive)
+RingHom.preserves1 (RingIso.ringHom (RingsIsomorphic.iso ringFirstIsomorphismTheorem')) = RingHom.preserves1 hom
+RingHom.ringHom (RingIso.ringHom (RingsIsomorphic.iso ringFirstIsomorphismTheorem')) {r} {s} = RingHom.ringHom hom
+RingHom.groupHom (RingIso.ringHom (RingsIsomorphic.iso ringFirstIsomorphismTheorem')) = GroupIso.groupHom (GroupsIsomorphic.proof (groupFirstIsomorphismTheorem'))
+RingIso.bijective (RingsIsomorphic.iso ringFirstIsomorphismTheorem') = GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem')
diff --git a/Rings/Ideals/Lemmas.agda b/Rings/Ideals/Lemmas.agda
new file mode 100644
index 0000000..d00a9b9
--- /dev/null
+++ b/Rings/Ideals/Lemmas.agda
@@ -0,0 +1,58 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+open import Groups.Subgroups.Definition
+open import Rings.Homomorphisms.Kernel
+open import Rings.Cosets
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Ideals.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
+
+open import Rings.Ideals.Definition R
+
+idealPredForKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → A → Set d
+idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
+
+idealPredForKernelWellDefined : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → {x y : A} → (Setoid._∼_ S x y) → (idealPredForKernel R2 fHom x → idealPredForKernel R2 fHom y)
+idealPredForKernelWellDefined {T = T} R2 {f} fHom a x=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined (RingHom.groupHom fHom) (Equivalence.symmetric (Setoid.eq S) a)) x=0
+
+kernelIdealIsIdeal : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} {R2 : Ring T _+2_ _*2_} {f : A → C} (fHom : RingHom R R2 f) → Ideal (idealPredForKernel R2 fHom)
+Subgroup.isSubset (Ideal.isSubgroup (kernelIdealIsIdeal {R2 = R2} fHom)) = idealPredForKernelWellDefined R2 fHom
+Subgroup.closedUnderPlus (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} {y} fx=0 fy=0 = transitive (transitive (GroupHom.groupHom (RingHom.groupHom fHom)) (+WellDefined fx=0 fy=0)) identLeft
+  where
+    open Ring R2
+    open Group (Ring.additiveGroup R2)
+    open Setoid T
+    open Equivalence eq
+Subgroup.containsIdentity (Ideal.isSubgroup (kernelIdealIsIdeal fHom)) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
+Subgroup.closedUnderInverse (Ideal.isSubgroup (kernelIdealIsIdeal {T = T} {R2 = R2} fHom)) {x} fx=0 = zeroImpliesInverseZero (RingHom.groupHom fHom) fx=0
+  where
+    open Ring R2
+    open Group (Ring.additiveGroup R2)
+    open Setoid T
+    open Equivalence eq
+Ideal.accumulatesTimes (kernelIdealIsIdeal {T = T} {R2 = R2} {f = f} fHom) {x} {y} fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2 {f y})))
+  where
+    open Setoid T
+    open Equivalence eq
+
+open Setoid S
+open Ring R
+open Group additiveGroup
+open Equivalence eq
+open import Groups.Lemmas additiveGroup
+
+idealIsKernelMap : {c : _} {pred : A → Set c} → (i : Ideal pred) → {x : A} → pred x → ringKernel {R1 = R} {R2 = cosetRing R i} (cosetRingHom R i)
+idealIsKernelMap {c} {pred} i {x} predX = x , (Ideal.isSubset i (transitive (symmetric identLeft) (+WellDefined (symmetric invIdent) reflexive)) predX)
diff --git a/Rings/Ideals/Prime/Definition.agda b/Rings/Ideals/Prime/Definition.agda
new file mode 100644
index 0000000..55128b2
--- /dev/null
+++ b/Rings/Ideals/Prime/Definition.agda
@@ -0,0 +1,22 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Lemmas
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Orders
+open import Setoids.Setoids
+open import Functions
+open import Rings.Definition
+open import Rings.Lemmas
+open import Sets.EquivalenceRelations
+open import Fields.Fields
+open import Rings.Ideals.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Ideals.Prime.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {pred : A → Set c} (i : Ideal R pred) where
+
+PrimeIdeal : Set (a ⊔ c)
+PrimeIdeal = {a b : A} → pred (a * b) → ((pred a) → False) → pred b
diff --git a/Rings/Ideals/Principal/Definition.agda b/Rings/Ideals/Principal/Definition.agda
new file mode 100644
index 0000000..d45d4bc
--- /dev/null
+++ b/Rings/Ideals/Principal/Definition.agda
@@ -0,0 +1,24 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Ideals.Principal.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
+
+open import Rings.Ideals.Definition R
+open Setoid S
+
+PrincipalIdeal : {c : _} {pred : A → Set c} (ideal : Ideal pred) → Set (a ⊔ b ⊔ c)
+PrincipalIdeal {pred = pred} ideal = Sg A (λ a → {x : A} → (pred x) → Sg A (λ c → (a * c) ∼ x))
diff --git a/Rings/IntegralDomains.agda b/Rings/IntegralDomains.agda
deleted file mode 100644
index 7cd2202..0000000
--- a/Rings/IntegralDomains.agda
+++ /dev/null
@@ -1,37 +0,0 @@
-{-# OPTIONS --safe --warning=error --without-K #-}
-
-open import LogicalFormulae
-open import Groups.Groups
-open import Groups.Lemmas
-open import Groups.Definition
-open import Numbers.Naturals.Naturals
-open import Orders
-open import Setoids.Setoids
-open import Functions
-open import Rings.Definition
-open import Rings.Lemmas
-open import Sets.EquivalenceRelations
-
-open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
-
-module Rings.IntegralDomains where
-  record IntegralDomain {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) : Set (lsuc m ⊔ n) where
-    field
-      intDom : {a b : A} → Setoid._∼_ S (a * b) (Ring.0R R) → (Setoid._∼_ S a (Ring.0R R)) || (Setoid._∼_ S b (Ring.0R R))
-      nontrivial : Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False
-
-  cancelIntDom : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) {a b c : A} → Setoid._∼_ S (a * b) (a * c) → (Setoid._∼_ S a (Ring.0R R)) || (Setoid._∼_ S b c)
-  cancelIntDom {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I {a} {b} {c} ab=ac = t4
-    where
-      open Setoid S
-      open Equivalence eq
-      t1 : (a * b) + Group.inverse (Ring.additiveGroup R) (a * c) ∼ Ring.0R R
-      t1 = transferToRight'' (Ring.additiveGroup R) ab=ac
-      t2 : a * (b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R
-      t2 = transitive (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) reflexive (transferToRight' (Ring.additiveGroup R) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (Ring.timesZero R)))))) t1
-      t3 : (a ∼ Ring.0R R) || ((b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R)
-      t3 = IntegralDomain.intDom I t2
-      t4 : (a ∼ Ring.0R R) || (b ∼ c)
-      t4 with t3
-      ... | inl t = inl t
-      ... | inr b = inr (transferToRight (Ring.additiveGroup R) b)
diff --git a/Rings/IntegralDomains/Definition.agda b/Rings/IntegralDomains/Definition.agda
new file mode 100644
index 0000000..ca4ccc4
--- /dev/null
+++ b/Rings/IntegralDomains/Definition.agda
@@ -0,0 +1,41 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Lemmas
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Orders
+open import Setoids.Setoids
+open import Functions
+open import Rings.Definition
+open import Rings.Lemmas
+open import Sets.EquivalenceRelations
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.IntegralDomains.Definition {m n : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
+
+open Setoid S
+open Equivalence eq
+open Ring R
+
+record IntegralDomain : Set (lsuc m ⊔ n) where
+  field
+    intDom : {a b : A} → (a * b) ∼ (Ring.0R R) → ((a ∼ (Ring.0R R)) → False) → b ∼ (Ring.0R R)
+    nontrivial : Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False
+
+decidedIntDom : ({a b : A} → (a * b) ∼ (Ring.0R R) → (a ∼ 0R) || (b ∼ 0R)) → ({a b : A} → (a * b) ∼ 0R → ((a ∼ (Ring.0R R)) → False) → b ∼ (Ring.0R R))
+decidedIntDom f ab=0 a!=0 with f ab=0
+decidedIntDom f ab=0 a!=0 | inl x = exFalso (a!=0 x)
+decidedIntDom f ab=0 a!=0 | inr x = x
+
+cancelIntDom : (I : IntegralDomain) {a b c : A} → (a * b) ∼ (a * c) → ((a ∼ (Ring.0R R)) → False) → (b ∼ c)
+cancelIntDom I {a} {b} {c} ab=ac a!=0 = transferToRight (Ring.additiveGroup R) t3
+  where
+    t1 : (a * b) + Group.inverse (Ring.additiveGroup R) (a * c) ∼ Ring.0R R
+    t1 = transferToRight'' (Ring.additiveGroup R) ab=ac
+    t2 : a * (b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R
+    t2 = transitive (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) reflexive (transferToRight' (Ring.additiveGroup R) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (Ring.timesZero R)))))) t1
+    t3 : (b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R
+    t3 = IntegralDomain.intDom I t2 a!=0
diff --git a/Rings/IntegralDomains/Examples.agda b/Rings/IntegralDomains/Examples.agda
new file mode 100644
index 0000000..32d0e22
--- /dev/null
+++ b/Rings/IntegralDomains/Examples.agda
@@ -0,0 +1,28 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Lemmas
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Orders
+open import Setoids.Setoids
+open import Functions
+open import Rings.Definition
+open import Rings.Lemmas
+open import Sets.EquivalenceRelations
+open import Rings.IntegralDomains.Definition
+open import Fields.Fields
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.IntegralDomains.Examples where
+
+fieldIsIntDom : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → (Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → IntegralDomain R
+IntegralDomain.intDom (fieldIsIntDom F 1!=0) {a} {b} ab=0 a!=0 with Field.allInvertible F a a!=0
+IntegralDomain.intDom (fieldIsIntDom {S = S} {R = R} F _) {a} {b} ab=0 a!=0 | 1/a , prA = transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric prA) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive ab=0) timesZero)))
+  where
+    open Setoid S
+    open Equivalence eq
+    open Ring R
+IntegralDomain.nontrivial (fieldIsIntDom F 1!=0) = 1!=0
diff --git a/Rings/Isomorphisms/Definition.agda b/Rings/Isomorphisms/Definition.agda
index af5f08d..1a2aa2e 100644
--- a/Rings/Isomorphisms/Definition.agda
+++ b/Rings/Isomorphisms/Definition.agda
@@ -19,4 +19,9 @@ module Rings.Isomorphisms.Definition {a b c d : _} {A : Set a} {S : Setoid {a} {
 record RingIso (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
   field
     ringHom : RingHom R1 R2 f
-    bijective : Bijection f
+    bijective : SetoidBijection S T f
+
+record RingsIsomorphic : Set (a ⊔ b ⊔ c ⊔ d) where
+  field
+    f : A → B
+    iso : RingIso f
diff --git a/Rings/PrincipalIdealDomain.agda b/Rings/PrincipalIdealDomain.agda
new file mode 100644
index 0000000..c399005
--- /dev/null
+++ b/Rings/PrincipalIdealDomain.agda
@@ -0,0 +1,24 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.PrincipalIdealDomain {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
+
+open import Rings.Ideals.Principal.Definition R
+open import Rings.Ideals.Definition R
+
+PrincipalIdealDomain : {c : _} → Set (a ⊔ b ⊔ lsuc c)
+PrincipalIdealDomain {c} = {pred : A → Set c} → (ideal : Ideal pred) → PrincipalIdeal ideal
diff --git a/Rings/Subrings/Definition.agda b/Rings/Subrings/Definition.agda
new file mode 100644
index 0000000..6b67b27
--- /dev/null
+++ b/Rings/Subrings/Definition.agda
@@ -0,0 +1,44 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Homomorphisms.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Subrings.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ _*A_ : A → A → A} (R : Ring S _+A_ _*A_) where
+
+open import Setoids.Subset S
+open Ring R
+open Group additiveGroup
+open import Groups.Subgroups.Definition additiveGroup
+open Setoid S
+open Equivalence eq
+
+record Subring {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
+  field
+    isSubgroup : Subgroup pred
+    containsOne : pred 1R
+    closedUnderProduct : {x y : A} → pred x → pred y → pred (x *A y)
+  isSubset = Subgroup.isSubset isSubgroup
+
+subringMult : {c : _} {pred : A → Set c} → (s : Subring pred) → Sg A pred → Sg A pred → Sg A pred
+subringMult s (a , prA) (b , prB) = (a *A b) , Subring.closedUnderProduct s prA prB
+
+subringIsRing : {c : _} {pred : A → Set c} → (subring : Subring pred) → Ring (subsetSetoid (Subring.isSubset subring)) (subgroupOp (Subring.isSubgroup subring)) (subringMult subring)
+Ring.additiveGroup (subringIsRing sub) = subgroupIsGroup (Subring.isSubgroup sub)
+Ring.*WellDefined (subringIsRing sub) {r , prR} {s , prS} {t , prT} {u , prU} r=t s=u = *WellDefined r=t s=u
+Ring.1R (subringIsRing sub) = (1R , Subring.containsOne sub)
+Ring.groupIsAbelian (subringIsRing sub) {a , prA} {b , prB} = groupIsAbelian
+Ring.*Associative (subringIsRing sub) {a , prA} {b , prB} {c , prC} = *Associative
+Ring.*Commutative (subringIsRing sub) {a , prA} {b , prB} = *Commutative
+Ring.*DistributesOver+ (subringIsRing sub) {a , prA} {b , prB} {c , prC} = *DistributesOver+
+Ring.identIsIdent (subringIsRing sub) {a , prA} = identIsIdent