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Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-qs 6401* Define quotient set.  R is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)
 |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
 
Theoremereq1 6402 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  =  S  ->  ( R  Er  A  <->  S  Er  A ) )
 
Theoremereq2 6403 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  =  B  ->  ( R  Er  A  <->  R  Er  B ) )
 
Theoremerrel 6404 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  Rel  R )
 
Theoremerdm 6405 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  dom  R  =  A )
 
Theoremercl 6406 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  A  e.  X )
 
Theoremersym 6407 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B R A )
 
Theoremercl2 6408 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  B  e.  X )
 
Theoremersymb 6409 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  ( A R B  <->  B R A ) )
 
Theoremertr 6410 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  (
 ( A R B  /\  B R C ) 
 ->  A R C ) )
 
Theoremertrd 6411 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr2d 6412 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  C R A )
 
Theoremertr3d 6413 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B R A )   &    |-  ( ph  ->  B R C )   =>    |-  ( ph  ->  A R C )
 
Theoremertr4d 6414 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   &    |-  ( ph  ->  C R B )   =>    |-  ( ph  ->  A R C )
 
Theoremerref 6415 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  A R A )
 
Theoremercnv 6416 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  `' R  =  R )
 
Theoremerrn 6417 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ran  R  =  A )
 
Theoremerssxp 6418 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  R  C_  ( A  X.  A ) )
 
Theoremerex 6419 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( A  e.  V  ->  R  e.  _V )
 )
 
Theoremerexb 6420 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( R  Er  A  ->  ( R  e.  _V  <->  A  e.  _V ) )
 
Theoremiserd 6421* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ( ph  /\  x R y )  ->  y R x )   &    |-  (
 ( ph  /\  ( x R y  /\  y R z ) ) 
 ->  x R z )   &    |-  ( ph  ->  ( x  e.  A  <->  x R x ) )   =>    |-  ( ph  ->  R  Er  A )
 
Theorembrdifun 6422 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B 
 <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
 
Theoremswoer 6423* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   =>    |-  ( ph  ->  R  Er  X )
 
Theoremswoord1 6424* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
 
Theoremswoord2 6425* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
 |-  R  =  ( ( X  X.  X ) 
 \  (  .<  u.  `'  .<  ) )   &    |-  ( ( ph  /\  ( y  e.  X  /\  z  e.  X ) )  ->  ( y 
 .<  z  ->  -.  z  .<  y ) )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x  .<  y  ->  ( x  .<  z  \/  z  .<  y )
 ) )   &    |-  ( ph  ->  B  e.  X )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
 
Theoremeqerlem 6426* Lemma for eqer 6427. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  ( z R w  <->  [_ z  /  x ]_ A  =  [_ w  /  x ]_ A )
 
Theoremeqer 6427* Equivalence relation involving equality of dependent classes  A
( x ) and  B ( y ). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( x  =  y 
 ->  A  =  B )   &    |-  R  =  { <. x ,  y >.  |  A  =  B }   =>    |-  R  Er  _V
 
Theoremider 6428 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
 |- 
 _I  Er  _V
 
Theorem0er 6429 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  (/)  Er  (/)
 
Theoremeceq1 6430 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
 
Theoremeceq1d 6431 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  [ A ] C  =  [ B ] C )
 
Theoremeceq2 6432 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  [ C ] A  =  [ C ] B )
 
Theoremelecg 6433 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremelec 6434 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  e.  [ B ] R  <->  B R A )
 
Theoremrelelec 6435 Membership in an equivalence class when  R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( Rel  R  ->  ( A  e.  [ B ] R  <->  B R A ) )
 
Theoremecss 6436 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   =>    |-  ( ph  ->  [ A ] R  C_  X )
 
Theoremecdmn0m 6437* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
 |-  ( A  e.  dom  R  <->  E. x  x  e.  [ A ] R )
 
Theoremereldm 6438 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  [ A ] R  =  [ B ] R )   =>    |-  ( ph  ->  ( A  e.  X  <->  B  e.  X ) )
 
Theoremerth 6439 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerth2 6440 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  B  e.  X )   =>    |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R ) )
 
Theoremerthi 6441 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  A R B )   =>    |-  ( ph  ->  [ A ] R  =  [ B ] R )
 
Theoremecidsn 6442 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)
 |- 
 [ A ]  _I  =  { A }
 
Theoremqseq1 6443 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( A /. C )  =  ( B /. C ) )
 
Theoremqseq2 6444 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( A  =  B  ->  ( C /. A )  =  ( C /. B ) )
 
Theoremelqsg 6445* Closed form of elqs 6446. (Contributed by Rodolfo Medina, 12-Oct-2010.)
 |-  ( B  e.  V  ->  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R ) )
 
Theoremelqs 6446* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  B  e.  _V   =>    |-  ( B  e.  ( A /. R )  <->  E. x  e.  A  B  =  [ x ] R )
 
Theoremelqsi 6447* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
 |-  ( B  e.  ( A /. R )  ->  E. x  e.  A  B  =  [ x ] R )
 
Theoremecelqsg 6448 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecelqsi 6449 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  R  e.  _V   =>    |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )
 
Theoremecopqsi 6450 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
 |-  R  e.  _V   &    |-  S  =  ( ( A  X.  A ) /. R )   =>    |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S )
 
Theoremqsexg 6451 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A /. R )  e.  _V )
 
Theoremqsex 6452 A quotient set exists. (Contributed by NM, 14-Aug-1995.)
 |-  A  e.  _V   =>    |-  ( A /. R )  e.  _V
 
Theoremuniqs 6453 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
 |-  ( R  e.  V  ->  U. ( A /. R )  =  ( R " A ) )
 
Theoremqsss 6454 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  A )   =>    |-  ( ph  ->  ( A /. R )  C_  ~P A )
 
Theoremuniqs2 6455 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
 |-  ( ph  ->  R  Er  A )   &    |-  ( ph  ->  R  e.  V )   =>    |-  ( ph  ->  U. ( A /. R )  =  A )
 
Theoremsnec 6456 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  { [ A ] R }  =  ( { A } /. R )
 
Theoremecqs 6457 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)
 |-  R  e.  _V   =>    |-  [ A ] R  =  U. ( { A } /. R )
 
Theoremecid 6458 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  A  e.  _V   =>    |-  [ A ] `'  _E  =  A
 
Theoremecidg 6459 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)
 |-  ( A  e.  V  ->  [ A ] `'  _E  =  A )
 
Theoremqsid 6460 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A /. `'  _E  )  =  A
 
Theoremectocld 6461* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 ( ch  /\  x  e.  B )  ->  ph )   =>    |-  (
 ( ch  /\  A  e.  S )  ->  ps )
 
Theoremectocl 6462* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  S  =  ( B
 /. R )   &    |-  ( [ x ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( x  e.  B  ->  ph )   =>    |-  ( A  e.  S  ->  ps )
 
Theoremelqsn0m 6463* An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  E. x  x  e.  B )
 
Theoremelqsn0 6464 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( dom  R  =  A  /\  B  e.  ( A /. R ) )  ->  B  =/=  (/) )
 
Theoremecelqsdm 6465 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)
 |-  ( ( dom  R  =  A  /\  [ B ] R  e.  ( A /. R ) ) 
 ->  B  e.  A )
 
Theoremxpider 6466 A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( A  X.  A )  Er  A
 
Theoremiinerm 6467* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  |^|_ x  e.  A  R  Er  B )
 
Theoremriinerm 6468* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  R  Er  B )  ->  (
 ( B  X.  B )  i^i  |^|_ x  e.  A  R )  Er  B )
 
Theoremerinxp 6469 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  R  Er  A )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ph  ->  ( R  i^i  ( B  X.  B ) )  Er  B )
 
Theoremecinxp 6470 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  ( ( ( R
 " A )  C_  A  /\  B  e.  A )  ->  [ B ] R  =  [ B ] ( R  i^i  ( A  X.  A ) ) )
 
Theoremqsinxp 6471 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ( R " A )  C_  A  ->  ( A /. R )  =  ( A /. ( R  i^i  ( A  X.  A ) ) ) )
 
Theoremqsel 6472 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)
 |-  ( ( R  Er  X  /\  B  e.  ( A /. R )  /\  C  e.  B )  ->  B  =  [ C ] R )
 
Theoremqliftlem 6473*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ( ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
 
Theoremqliftrel 6474*  F, a function lift, is a subset of  R  X.  S. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  F  C_  ( ( X /. R )  X.  Y ) )
 
Theoremqliftel 6475* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( [ C ] R F D 
 <-> 
 E. x  e.  X  ( C R x  /\  D  =  A )
 ) )
 
Theoremqliftel1 6476* Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ( ph  /\  x  e.  X )  ->  [ x ] R F A )
 
Theoremqliftfun 6477* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  y 
 ->  A  =  B )   =>    |-  ( ph  ->  ( Fun  F  <->  A. x A. y ( x R y  ->  A  =  B )
 ) )
 
Theoremqliftfund 6478* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  y 
 ->  A  =  B )   &    |-  ( ( ph  /\  x R y )  ->  A  =  B )   =>    |-  ( ph  ->  Fun  F )
 
Theoremqliftfuns 6479* The function  F is the unique function defined by  F `  [
x ]  =  A, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( Fun  F  <->  A. y A. z
 ( y R z 
 ->  [_ y  /  x ]_ A  =  [_ z  /  x ]_ A ) ) )
 
Theoremqliftf 6480* The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   =>    |-  ( ph  ->  ( Fun  F  <->  F : ( X
 /. R ) --> Y ) )
 
Theoremqliftval 6481* The value of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )   &    |-  ( ( ph  /\  x  e.  X )  ->  A  e.  Y )   &    |-  ( ph  ->  R  Er  X )   &    |-  ( ph  ->  X  e.  _V )   &    |-  ( x  =  C  ->  A  =  B )   &    |-  ( ph  ->  Fun  F )   =>    |-  ( ( ph  /\  C  e.  X )  ->  ( F `  [ C ] R )  =  B )
 
Theoremecoptocl 6482* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
 |-  S  =  ( ( B  X.  C )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  (
 ( x  e.  B  /\  y  e.  C )  ->  ph )   =>    |-  ( A  e.  S  ->  ps )
 
Theorem2ecoptocl 6483* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)
 |-  S  =  ( ( C  X.  D )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( [ <. z ,  w >. ] R  =  B  ->  ( ps  <->  ch ) )   &    |-  (
 ( ( x  e.  C  /\  y  e.  D )  /\  (
 z  e.  C  /\  w  e.  D )
 )  ->  ph )   =>    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ch )
 
Theorem3ecoptocl 6484* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)
 |-  S  =  ( ( D  X.  D )
 /. R )   &    |-  ( [ <. x ,  y >. ] R  =  A  ->  ( ph  <->  ps ) )   &    |-  ( [ <. z ,  w >. ] R  =  B  ->  ( ps  <->  ch ) )   &    |-  ( [ <. v ,  u >. ] R  =  C  ->  ( ch  <->  th ) )   &    |-  (
 ( ( x  e.  D  /\  y  e.  D )  /\  (
 z  e.  D  /\  w  e.  D )  /\  ( v  e.  D  /\  u  e.  D ) )  ->  ph )   =>    |-  (
 ( A  e.  S  /\  B  e.  S  /\  C  e.  S )  ->  th )
 
Theorembrecop 6485* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( G  X.  G )   &    |-  H  =  ( ( G  X.  G ) /.  .~  )   &    |- 
 .<_  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ] 
 .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }   &    |-  (
 ( ( ( z  e.  G  /\  w  e.  G )  /\  ( A  e.  G  /\  B  e.  G )
 )  /\  ( (
 v  e.  G  /\  u  e.  G )  /\  ( C  e.  G  /\  D  e.  G ) ) )  ->  (
 ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ] 
 .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps ) ) )   =>    |-  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
 )  ->  ( [ <. A ,  B >. ] 
 .~  .<_  [ <. C ,  D >. ]  .~  <->  ps ) )
 
Theoremeroveu 6486* Lemma for eroprf 6488. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   =>    |-  ( ( ph  /\  ( X  e.  J  /\  Y  e.  K )
 )  ->  E! z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T ) )
 
Theoremerovlem 6487* Lemma for eroprf 6488. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   &    |-  .+^ 
 =  { <. <. x ,  y >. ,  z >.  | 
 E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) }   =>    |-  ( ph  ->  .+^  =  ( x  e.  J ,  y  e.  K  |->  ( iota
 z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) ) ) )
 
Theoremeroprf 6488* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
 |-  J  =  ( A
 /. R )   &    |-  K  =  ( B /. S )   &    |-  ( ph  ->  T  e.  Z )   &    |-  ( ph  ->  R  Er  U )   &    |-  ( ph  ->  S  Er  V )   &    |-  ( ph  ->  T  Er  W )   &    |-  ( ph  ->  A 
 C_  U )   &    |-  ( ph  ->  B  C_  V )   &    |-  ( ph  ->  C  C_  W )   &    |-  ( ph  ->  .+ 
 : ( A  X.  B ) --> C )   &    |-  ( ( ph  /\  (
 ( r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B ) ) ) 
 ->  ( ( r R s  /\  t S u )  ->  (
 r  .+  t ) T ( s  .+  u ) ) )   &    |-  .+^ 
 =  { <. <. x ,  y >. ,  z >.  | 
 E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
 q ] S ) 
 /\  z  =  [
 ( p  .+  q
 ) ] T ) }   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  S  e.  Y )   &    |-  L  =  ( C
 /. T )   =>    |-  ( ph  ->  .+^  : ( J  X.  K )
 --> L )
 
Theoremeroprf2 6489* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  J  =  ( A
 /.  .~  )   &    |-  .+^  =  { <.
 <. x ,  y >. ,  z >.  |  E. p  e.  A  E. q  e.  A  ( ( x  =  [ p ]  .~  /\  y  =  [
 q ]  .~  )  /\  z  =  [
 ( p  .+  q
 ) ]  .~  ) }   &    |-  ( ph  ->  .~  e.  X )   &    |-  ( ph  ->  .~ 
 Er  U )   &    |-  ( ph  ->  A  C_  U )   &    |-  ( ph  ->  .+  :
 ( A  X.  A )
 --> A )   &    |-  ( ( ph  /\  ( ( r  e.  A  /\  s  e.  A )  /\  (
 t  e.  A  /\  u  e.  A )
 ) )  ->  (
 ( r  .~  s  /\  t  .~  u ) 
 ->  ( r  .+  t
 )  .~  ( s  .+  u ) ) )   =>    |-  ( ph  ->  .+^  : ( J  X.  J ) --> J )
 
Theoremecopoveq 6490* This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation 
.~ (specified by the hypothesis) in terms of its operation  F. (Contributed by NM, 16-Aug-1995.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   =>    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( <. A ,  B >.  .~  <. C ,  D >.  <->  ( A  .+  D )  =  ( B  .+  C ) ) )
 
Theoremecopovsym 6491* Assuming the operation  F is commutative, show that the relation  .~, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( x  .+  y )  =  (
 y  .+  x )   =>    |-  ( A  .~  B  ->  B  .~  A )
 
Theoremecopovtrn 6492* Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( x  .+  y )  =  (
 y  .+  x )   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |-  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C )
 
Theoremecopover 6493* Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( x  .+  y )  =  (
 y  .+  x )   &    |-  (
 ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  .+  y )  .+  z )  =  ( x  .+  ( y  .+  z ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |- 
 .~  Er  ( S  X.  S )
 
Theoremecopovsymg 6494* Assuming the operation  F is commutative, show that the relation  .~, specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  =  ( y  .+  x ) )   =>    |-  ( A  .~  B  ->  B  .~  A )
 
Theoremecopovtrng 6495* Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is transitive. (Contributed by Jim Kingdon, 1-Sep-2019.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |-  ( ( A  .~  B  /\  B  .~  C )  ->  A  .~  C )
 
Theoremecopoverg 6496* Assuming that operation  F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation  .~, specified by the first hypothesis, is an equivalence relation. (Contributed by Jim Kingdon, 1-Sep-2019.)
 |- 
 .~  =  { <. x ,  y >.  |  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  .+  u )  =  ( w  .+  v ) ) ) }   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  =  ( y  .+  x ) )   &    |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x  .+  y )  e.  S )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  (
 ( x  .+  y
 )  =  ( x 
 .+  z )  ->  y  =  z )
 )   =>    |- 
 .~  Er  ( S  X.  S )
 
Theoremth3qlem1 6497* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |- 
 .~  Er  S   &    |-  ( ( ( y  e.  S  /\  w  e.  S )  /\  ( z  e.  S  /\  v  e.  S ) )  ->  ( ( y  .~  w  /\  z  .~  v )  ->  ( y  .+  z ) 
 .~  ( w  .+  v ) ) )   =>    |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S
 /.  .~  ) )  ->  E* x E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [
 z ]  .~  )  /\  x  =  [
 ( y  .+  z
 ) ]  .~  )
 )
 
Theoremth3qlem2 6498* Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( S  X.  S )   &    |-  (
 ( ( ( w  e.  S  /\  v  e.  S )  /\  ( u  e.  S  /\  t  e.  S )
 )  /\  ( (
 s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S ) ) )  ->  ( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+ 
 <. s ,  f >. ) 
 .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )   =>    |-  ( ( A  e.  ( ( S  X.  S ) /.  .~  )  /\  B  e.  (
 ( S  X.  S ) /.  .~  ) ) 
 ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
 ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  )
 )
 
Theoremth3qcor 6499* Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( S  X.  S )   &    |-  (
 ( ( ( w  e.  S  /\  v  e.  S )  /\  ( u  e.  S  /\  t  e.  S )
 )  /\  ( (
 s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S ) ) )  ->  ( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+ 
 <. s ,  f >. ) 
 .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )   &    |-  G  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  (
 ( S  X.  S ) /.  .~  ) ) 
 /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ] 
 .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+ 
 <. u ,  t >. ) ]  .~  ) ) }   =>    |- 
 Fun  G
 
Theoremth3q 6500* Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |- 
 .~  e.  _V   &    |-  .~  Er  ( S  X.  S )   &    |-  (
 ( ( ( w  e.  S  /\  v  e.  S )  /\  ( u  e.  S  /\  t  e.  S )
 )  /\  ( (
 s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S ) ) )  ->  ( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+ 
 <. s ,  f >. ) 
 .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )   &    |-  G  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  (
 ( S  X.  S ) /.  .~  ) ) 
 /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ] 
 .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+ 
 <. u ,  t >. ) ]  .~  ) ) }   =>    |-  ( ( ( A  e.  S  /\  B  e.  S )  /\  ( C  e.  S  /\  D  e.  S )
 )  ->  ( [ <. A ,  B >. ] 
 .~  G [ <. C ,  D >. ]  .~  )  =  [ ( <. A ,  B >.  .+ 
 <. C ,  D >. ) ]  .~  )
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