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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeleei 23901 The forward direction of elee 23898. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ...
 N ) --> RR )
 
Theoremeedimeq 23902 A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  A  e.  ( EE `  M ) )  ->  N  =  M )
 
Theorembrbtwn 23903* The binary relationship form of the betweenness predicate. The statement  A  Btwn  <. B ,  C >. should be informally read as " A lies on a line segment between  B and  C. This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A 
 Btwn  <. B ,  C >.  <->  E. t  e.  (
 0 [,] 1 ) A. i  e.  ( 1 ... N ) ( A `
  i )  =  ( ( ( 1  -  t )  x.  ( B `  i
 ) )  +  (
 t  x.  ( C `
  i ) ) ) ) )
 
Theorembrcgr 23904* The binary relationship form of the congruence predicate. The statement  <. A ,  B >.Cgr <. C ,  D >. should be read informally as "the  N dimensional point  A is as far from  B as  C is from  D, or "the line segment  A B is congruent to the line segment  C D. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. C ,  D >. 
 <-> 
 sum_ i  e.  (
 1 ... N ) ( ( ( A `  i )  -  ( B `  i ) ) ^ 2 )  = 
 sum_ i  e.  (
 1 ... N ) ( ( ( C `  i )  -  ( D `  i ) ) ^ 2 ) ) )
 
Theoremfveere 23905 The function value of a point is a real. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  I  e.  ( 1 ... N ) )  ->  ( A `  I )  e.  RR )
 
Theoremfveecn 23906 The function value of a point is a complex. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  I  e.  ( 1 ... N ) )  ->  ( A `  I )  e.  CC )
 
Theoremeqeefv 23907* Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( A  =  B  <->  A. i  e.  ( 1
 ... N ) ( A `  i )  =  ( B `  i ) ) )
 
Theoremeqeelen 23908* Two points are equal iff the square of the distance between them is zero. (Contributed by Scott Fenton, 10-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( A  =  B  <->  sum_
 i  e.  ( 1
 ... N ) ( ( ( A `  i )  -  ( B `  i ) ) ^ 2 )  =  0 ) )
 
Theorembrbtwn2 23909* Alternate characterization of betweenness, with no existential quantifiers. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A 
 Btwn  <. B ,  C >.  <-> 
 ( A. i  e.  (
 1 ... N ) ( ( ( B `  i )  -  ( A `  i ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <_  0  /\  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) ) ) ) )
 
Theoremcolinearalglem1 23910 Lemma for colinearalg 23914. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  E  e.  CC  /\  F  e.  CC )
 )  ->  ( (
 ( B  -  A )  x.  ( F  -  D ) )  =  ( ( E  -  D )  x.  ( C  -  A ) )  <-> 
 ( ( B  x.  F )  -  (
 ( A  x.  F )  +  ( B  x.  D ) ) )  =  ( ( C  x.  E )  -  ( ( A  x.  E )  +  ( C  x.  D ) ) ) ) )
 
Theoremcolinearalglem2 23911* Lemma for colinearalg 23914. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A. i  e.  ( 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( A `  i ) )  x.  ( ( C `  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j
 )  -  ( A `
  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( C `  i
 )  -  ( B `
  i ) )  x.  ( ( A `
  j )  -  ( B `  j ) ) )  =  ( ( ( C `  j )  -  ( B `  j ) )  x.  ( ( A `
  i )  -  ( B `  i ) ) ) ) )
 
Theoremcolinearalglem3 23912* Lemma for colinearalg 23914. Translate between two forms of the colinearity condition. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( A. i  e.  ( 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( A `  i ) )  x.  ( ( C `  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j
 )  -  ( A `
  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( A `  i
 )  -  ( C `
  i ) )  x.  ( ( B `
  j )  -  ( C `  j ) ) )  =  ( ( ( A `  j )  -  ( C `  j ) )  x.  ( ( B `
  i )  -  ( C `  i ) ) ) ) )
 
Theoremcolinearalglem4 23913* Lemma for colinearalg 23914. Prove a disjunction that will be needed in the final proof. (Contributed by Scott Fenton, 27-Jun-2013.)
 |-  (
 ( ( A  e.  ( EE `  N ) 
 /\  C  e.  ( EE `  N ) ) 
 /\  K  e.  RR )  ->  ( A. i  e.  ( 1 ... N ) ( ( ( ( K  x.  (
 ( C `  i
 )  -  ( A `
  i ) ) )  +  ( A `
  i ) )  -  ( A `  i ) )  x.  ( ( C `  i )  -  ( A `  i ) ) )  <_  0  \/  A. i  e.  ( 1
 ... N ) ( ( ( C `  i )  -  (
 ( K  x.  (
 ( C `  i
 )  -  ( A `
  i ) ) )  +  ( A `
  i ) ) )  x.  ( ( A `  i )  -  ( ( K  x.  ( ( C `
  i )  -  ( A `  i ) ) )  +  ( A `  i ) ) ) )  <_  0  \/  A. i  e.  (
 1 ... N ) ( ( ( A `  i )  -  ( C `  i ) )  x.  ( ( ( K  x.  ( ( C `  i )  -  ( A `  i ) ) )  +  ( A `  i ) )  -  ( C `  i ) ) )  <_  0
 ) )
 
Theoremcolinearalg 23914* An algebraic characterization of colinearity. Note the similarity to brbtwn2 23909. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  (
 ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  ->  ( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  <->  A. i  e.  (
 1 ... N ) A. j  e.  ( 1 ... N ) ( ( ( B `  i
 )  -  ( A `
  i ) )  x.  ( ( C `
  j )  -  ( A `  j ) ) )  =  ( ( ( B `  j )  -  ( A `  j ) )  x.  ( ( C `
  i )  -  ( A `  i ) ) ) ) )
 
Theoremeleesub 23915* Membership of a subtraction mapping in a Euclidean space. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  C  =  ( i  e.  (
 1 ... N )  |->  ( ( A `  i
 )  -  ( B `
  i ) ) )   =>    |-  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 ->  C  e.  ( EE
 `  N ) )
 
Theoremeleesubd 23916* Membership of a subtraction mapping in a Euclidean space. Deduction form of eleesub 23915. (Contributed by Scott Fenton, 17-Jul-2013.)
 |-  ( ph  ->  C  =  ( i  e.  ( 1
 ... N )  |->  ( ( A `  i
 )  -  ( B `
  i ) ) ) )   =>    |-  ( ( ph  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) 
 ->  C  e.  ( EE
 `  N ) )
 
18.7.31  Tarski's axioms for geometry
 
Theoremaxdimuniq 23917 The unique dimensional axiom. If a point is in  N dimensional space and in  M dimensional space, then  N  =  M. This axiom is not traditionally presented with Tarski's axioms, but we require it here as we are considering spaces in arbitrary dimensions. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N ) ) 
 /\  ( M  e.  NN  /\  A  e.  ( EE `  M ) ) )  ->  N  =  M )
 
Theoremaxcgrrflx 23918  A is as far from  B as  B is from  A. Axiom A1 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. B ,  A >. )
 
Theoremaxcgrtr 23919 Congruence is transitive. Axiom A2 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.Cgr <. C ,  D >.  /\ 
 <. A ,  B >.Cgr <. E ,  F >. ) 
 ->  <. C ,  D >.Cgr
 <. E ,  F >. ) )
 
Theoremaxcgrid 23920 If there is no distance between  A and  B, then  A  =  B. Axiom A3 of [Schwabhauser] p. 10. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr
 <. C ,  C >.  ->  A  =  B )
 )
 
Theoremaxsegconlem1 23921* Lemma for axsegcon 23931. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)
 |-  (
 ( A  =  B  /\  ( ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N ) ) 
 /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) ) )  ->  E. x  e.  ( EE `  N ) E. t  e.  (
 0 [,] 1 ) (
 A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  t )  x.  ( A `  i ) )  +  ( t  x.  ( x `  i ) ) )  /\  sum_ i  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( x `  i ) ) ^
 2 )  =  sum_ i  e.  ( 1 ...
 N ) ( ( ( C `  i
 )  -  ( D `
  i ) ) ^ 2 ) ) )
 
Theoremaxsegconlem2 23922* Lemma for axsegcon 23931. Show that the square of the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  S  e.  RR )
 
Theoremaxsegconlem3 23923* Lemma for axsegcon 23931. Show that the square of the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  0  <_  S )
 
Theoremaxsegconlem4 23924* Lemma for axsegcon 23931. Show that the distance between two points is a real number. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  ( sqr `  S )  e.  RR )
 
Theoremaxsegconlem5 23925* Lemma for axsegcon 23931. Show that the distance between two points is non-negative. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  ->  0  <_  ( sqr `  S )
 )
 
Theoremaxsegconlem6 23926* Lemma for axsegcon 23931. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   =>    |-  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  ->  0  <  ( sqr `  S ) )
 
Theoremaxsegconlem7 23927* Lemma for axsegcon 23931. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   =>    |-  ( ( ( A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( sqr `  S )  /  ( ( sqr `  S )  +  ( sqr `  T ) ) )  e.  ( 0 [,] 1 ) )
 
Theoremaxsegconlem8 23928* Lemma for axsegcon 23931. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   &    |-  F  =  ( k  e.  (
 1 ... N )  |->  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `  k ) )  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S ) ) )   =>    |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N ) )
 
Theoremaxsegconlem9 23929* Lemma for axsegcon 23931. Show that  B F is congruent to  C D. (Contributed by Scott Fenton, 19-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   &    |-  F  =  ( k  e.  (
 1 ... N )  |->  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `  k ) )  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S ) ) )   =>    |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( B `  i )  -  ( F `  i ) ) ^
 2 )  =  sum_ i  e.  ( 1 ...
 N ) ( ( ( C `  i
 )  -  ( D `
  i ) ) ^ 2 ) )
 
Theoremaxsegconlem10 23930* Lemma for axsegcon 23931. Show that the scaling constant from axsegconlem7 23927 produces the betweenness condition for  A,  B and  F. (Contributed by Scott Fenton, 21-Sep-2013.)
 |-  S  =  sum_ p  e.  (
 1 ... N ) ( ( ( A `  p )  -  ( B `  p ) ) ^ 2 )   &    |-  T  =  sum_ p  e.  (
 1 ... N ) ( ( ( C `  p )  -  ( D `  p ) ) ^ 2 )   &    |-  F  =  ( k  e.  (
 1 ... N )  |->  ( ( ( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `  k ) )  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S ) ) )   =>    |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  A  =/=  B )  /\  ( C  e.  ( EE `  N ) 
 /\  D  e.  ( EE `  N ) ) )  ->  A. i  e.  ( 1 ... N ) ( B `  i )  =  (
 ( ( 1  -  ( ( sqr `  S )  /  ( ( sqr `  S )  +  ( sqr `  T ) ) ) )  x.  ( A `  i ) )  +  ( ( ( sqr `  S )  /  ( ( sqr `  S )  +  ( sqr `  T ) ) )  x.  ( F `  i ) ) ) )
 
Theoremaxsegcon 23931* Any segment  A B can be extended to a point  x such that  B x is congruent to  C D. Axiom A4 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 4-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  E. x  e.  ( EE `  N ) ( B  Btwn  <. A ,  x >.  /\  <. B ,  x >.Cgr <. C ,  D >. ) )
 
Theoremax5seglem1 23932* Lemma for ax5seg 23942. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1
 ... N ) ( ( ( A `  j )  -  ( B `  j ) ) ^ 2 )  =  ( ( T ^
 2 )  x.  sum_ j  e.  ( 1 ...
 N ) ( ( ( A `  j
 )  -  ( C `
  j ) ) ^ 2 ) ) )
 
Theoremax5seglem2 23933* Lemma for ax5seg 23942. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1
 ... N ) ( ( ( B `  j )  -  ( C `  j ) ) ^ 2 )  =  ( ( ( 1  -  T ) ^
 2 )  x.  sum_ j  e.  ( 1 ...
 N ) ( ( ( A `  j
 )  -  ( C `
  j ) ) ^ 2 ) ) )
 
Theoremax5seglem3a 23934 Lemma for ax5seg 23942. (Contributed by Scott Fenton, 7-May-2015.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  /\  j  e.  ( 1 ... N ) )  ->  ( ( ( A `
  j )  -  ( C `  j ) )  e.  RR  /\  ( ( D `  j )  -  ( F `  j ) )  e.  RR ) )
 
Theoremax5seglem3 23935* Lemma for ax5seg 23942. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) )  /\  ( ( T  e.  ( 0 [,] 1
 )  /\  S  e.  ( 0 [,] 1
 ) )  /\  ( A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  (
 ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i ) ) ) ) ) 
 /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  sum_ j  e.  ( 1 ... N ) ( ( ( A `  j )  -  ( C `  j ) ) ^
 2 )  =  sum_ j  e.  ( 1 ...
 N ) ( ( ( D `  j
 )  -  ( F `
  j ) ) ^ 2 ) )
 
Theoremax5seglem4 23936* Lemma for ax5seg 23942. Given two distinct points, the scaling constant in a betweenness statement is non-zero. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 /\  A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) )  /\  A  =/=  B )  ->  T  =/=  0 )
 
Theoremax5seglem5 23937* Lemma for ax5seg 23942. If  B is between  A and  C, and  A is distinct from  B, then  A is distinct from  C. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  e.  ( EE `  N ) 
 /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) 
 /\  ( A  =/=  B 
 /\  T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  sum_ j  e.  ( 1
 ... N ) ( ( ( A `  j )  -  ( C `  j ) ) ^ 2 )  =/=  0 )
 
Theoremax5seglem6 23938* Lemma for ax5seg 23942. Given two line segments that are divided into pieces, if the pieces are congruent, then the scaling constant is the same. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) 
 /\  C  e.  ( EE `  N ) ) 
 /\  ( D  e.  ( EE `  N ) 
 /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N ) ) ) )  /\  ( A  =/=  B  /\  ( T  e.  ( 0 [,] 1 )  /\  S  e.  ( 0 [,] 1
 ) )  /\  ( A. i  e.  (
 1 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) )  /\  A. i  e.  ( 1 ... N ) ( E `  i )  =  (
 ( ( 1  -  S )  x.  ( D `  i ) )  +  ( S  x.  ( F `  i ) ) ) ) ) 
 /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
 <. E ,  F >. ) )  ->  T  =  S )
 
Theoremax5seglem7 23939 Lemma for ax5seg 23942. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  A  e.  CC   &    |-  T  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( T  x.  ( ( C  -  D ) ^
 2 ) )  =  ( ( ( ( ( ( 1  -  T )  x.  A )  +  ( T  x.  C ) )  -  D ) ^ 2
 )  +  ( ( 1  -  T )  x.  ( ( T  x.  ( ( A  -  C ) ^
 2 ) )  -  ( ( A  -  D ) ^ 2
 ) ) ) )
 
Theoremax5seglem8 23940 Lemma for ax5seg 23942. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 23939. (Contributed by Scott Fenton, 11-Jun-2013.)
 |-  (
 ( ( A  e.  CC  /\  T  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( T  x.  ( ( C  -  D ) ^
 2 ) )  =  ( ( ( ( ( ( 1  -  T )  x.  A )  +  ( T  x.  C ) )  -  D ) ^ 2
 )  +  ( ( 1  -  T )  x.  ( ( T  x.  ( ( A  -  C ) ^
 2 ) )  -  ( ( A  -  D ) ^ 2
 ) ) ) ) )
 
Theoremax5seglem9 23941* Lemma for ax5seg 23942. Take the calculation in ax5seglem8 23940 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)
 |-  (
 ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) )  /\  ( T  e.  (
 0 [,] 1 )  /\  A. i  e.  ( 1
 ... N ) ( B `  i )  =  ( ( ( 1  -  T )  x.  ( A `  i ) )  +  ( T  x.  ( C `  i ) ) ) ) )  ->  ( T  x.  sum_ j  e.  ( 1 ... N ) ( ( ( C `  j )  -  ( D `  j ) ) ^
 2 ) )  =  ( sum_ j  e.  (
 1 ... N ) ( ( ( B `  j )  -  ( D `  j ) ) ^ 2 )  +  ( ( 1  -  T )  x.  (
 ( T  x.  sum_ j  e.  ( 1 ...
 N ) ( ( ( A `  j
 )  -  ( C `
  j ) ) ^ 2 ) )  -  sum_ j  e.  (
 1 ... N ) ( ( ( A `  j )  -  ( D `  j ) ) ^ 2 ) ) ) ) )
 
Theoremax5seg 23942 The five segment axiom. Take two triangles  A D C and  E H G, a point  B on  A C, and a point  F on  E G. If all corresponding line segments except for  C D and  G H are congruent, then so are  C D and  G H. Axiom A5 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  G  e.  ( EE `  N ) 
 /\  H  e.  ( EE `  N ) ) )  ->  ( (
 ( A  =/=  B  /\  B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. ) 
 /\  ( <. A ,  B >.Cgr <. E ,  F >.  /\  <. B ,  C >.Cgr
 <. F ,  G >. ) 
 /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr
 <. F ,  H >. ) )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
 
Theoremaxbtwnid 23943 Points are indivisible. That is, if  A lies between  B and  B, then  A  =  B. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  ( A  Btwn  <. B ,  B >.  ->  A  =  B ) )
 
Theoremaxpaschlem 23944* Lemma for axpasch 23945. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)
 |-  (
 ( T  e.  (
 0 [,] 1 )  /\  S  e.  ( 0 [,] 1 ) )  ->  E. r  e.  (
 0 [,] 1 ) E. p  e.  ( 0 [,] 1 ) ( p  =  ( ( 1  -  r )  x.  ( 1  -  T ) )  /\  r  =  ( ( 1  -  p )  x.  (
 1  -  S ) )  /\  ( ( 1  -  r )  x.  T )  =  ( ( 1  -  p )  x.  S ) ) )
 
Theoremaxpasch 23945* The inner Pasch axiom. Take a triangle  A C E, a point  D on  A C, and a point  B extending  C E. Then  A E and  D B intersect at some point  x. Axiom A7 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 3-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  (
 ( D  Btwn  <. A ,  C >.  /\  E  Btwn  <. B ,  C >. ) 
 ->  E. x  e.  ( EE `  N ) ( x  Btwn  <. D ,  B >.  /\  x  Btwn  <. E ,  A >. ) ) )
 
Theoremaxlowdimlem1 23946 Lemma for axlowdim 23965. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 3 ... N )  X.  { 0 } ) : ( 3
 ... N ) --> RR
 
Theoremaxlowdimlem2 23947 Lemma for axlowdim 23965. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  (
 ( 1 ... 2
 )  i^i  ( 3 ... N ) )  =  (/)
 
Theoremaxlowdimlem3 23948 Lemma for axlowdim 23965. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  ( 1
 ... N )  =  ( ( 1 ... 2 )  u.  (
 3 ... N ) ) )
 
Theoremaxlowdimlem4 23949 Lemma for axlowdim 23965. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  { <. 1 ,  A >. ,  <. 2 ,  B >. } :
 ( 1 ... 2
 ) --> RR
 
Theoremaxlowdimlem5 23950 Lemma for axlowdim 23965. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( N  e.  ( ZZ>= `  2 )  ->  ( { <. 1 ,  A >. , 
 <. 2 ,  B >. }  u.  ( ( 3
 ... N )  X.  { 0 } ) )  e.  ( EE `  N ) )
 
Theoremaxlowdimlem6 23951 Lemma for axlowdim 23965. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  A  =  ( { <. 1 ,  0 >. ,  <. 2 ,  0 >. }  u.  ( ( 3 ...
 N )  X.  {
 0 } ) )   &    |-  B  =  ( { <. 1 ,  1 >. ,  <. 2 ,  0
 >. }  u.  ( ( 3 ... N )  X.  { 0 } ) )   &    |-  C  =  ( { <. 1 ,  0
 >. ,  <. 2 ,  1
 >. }  u.  ( ( 3 ... N )  X.  { 0 } ) )   =>    |-  ( N  e.  ( ZZ>=
 `  2 )  ->  -.  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )
 
Theoremaxlowdimlem7 23952 Lemma for axlowdim 23965. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( N  e.  ( ZZ>=
 `  3 )  ->  P  e.  ( EE `  N ) )
 
Theoremaxlowdimlem8 23953 Lemma for axlowdim 23965. Calulate the value of  P at three. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( P `  3
 )  =  -u 1
 
Theoremaxlowdimlem9 23954 Lemma for axlowdim 23965. Calulate the value of  P away from three. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   =>    |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  3
 )  ->  ( P `  K )  =  0 )
 
Theoremaxlowdimlem10 23955 Lemma for axlowdim 23965. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   =>    |-  ( ( N  e.  NN  /\  I  e.  (
 1 ... ( N  -  1 ) ) ) 
 ->  Q  e.  ( EE
 `  N ) )
 
Theoremaxlowdimlem11 23956 Lemma for axlowdim 23965. Calculate the value of  Q at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   =>    |-  ( Q `  ( I  +  1 ) )  =  1
 
Theoremaxlowdimlem12 23957 Lemma for axlowdim 23965. Calculate the value of  Q away from its distunguished point. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   =>    |-  ( ( K  e.  ( 1 ... N )  /\  K  =/=  ( I  +  1 )
 )  ->  ( Q `  K )  =  0 )
 
Theoremaxlowdimlem13 23958 Lemma for axlowdim 23965. Establish that  P and 
Q are different points. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   &    |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( I  +  1 ) }
 )  X.  { 0 } ) )   =>    |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  -  1 ) ) )  ->  P  =/=  Q )
 
Theoremaxlowdimlem14 23959 Lemma for axlowdim 23965. Take two possible  Q from axlowdimlem10 23955. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  (
 ( ( 1 ...
 N )  \  {
 ( I  +  1 ) } )  X.  { 0 } ) )   &    |-  R  =  ( { <. ( J  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( J  +  1 ) }
 )  X.  { 0 } ) )   =>    |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  -  1 ) ) 
 /\  J  e.  (
 1 ... ( N  -  1 ) ) ) 
 ->  ( Q  =  R  ->  I  =  J ) )
 
Theoremaxlowdimlem15 23960* Lemma for axlowdim 23965. Set up a one to one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  F  =  ( i  e.  (
 1 ... ( N  -  1 ) )  |->  if ( i  =  1 ,  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 ) ,  ( { <. ( i  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( i  +  1 ) }
 )  X.  { 0 } ) ) ) )   =>    |-  ( N  e.  ( ZZ>=
 `  3 )  ->  F : ( 1 ... ( N  -  1
 ) ) -1-1-> ( EE
 `  N ) )
 
Theoremaxlowdimlem16 23961* Lemma for axlowdim 23965. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   &    |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( I  +  1 ) }
 )  X.  { 0 } ) )   =>    |-  ( ( N  e.  ( ZZ>= `  3
 )  /\  I  e.  ( 2 ... ( N  -  1 ) ) )  ->  sum_ i  e.  ( 3 ... N ) ( ( P `
  i ) ^
 2 )  =  sum_ i  e.  ( 3 ...
 N ) ( ( Q `  i ) ^ 2 ) )
 
Theoremaxlowdimlem17 23962 Lemma for axlowdim 23965. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)
 |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1
 ... N )  \  { 3 } )  X.  { 0 } )
 )   &    |-  Q  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N ) 
 \  { ( I  +  1 ) }
 )  X.  { 0 } ) )   &    |-  A  =  ( { <. 1 ,  X >. ,  <. 2 ,  Y >. }  u.  (
 ( 3 ... N )  X.  { 0 } ) )   &    |-  X  e.  RR   &    |-  Y  e.  RR   =>    |-  ( ( N  e.  ( ZZ>= `  3 )  /\  I  e.  (
 2 ... ( N  -  1 ) ) ) 
 ->  <. P ,  A >.Cgr
 <. Q ,  A >. )
 
Theoremaxlowdim1 23963* The lower dimensional axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 23964. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  NN  ->  E. x  e.  ( EE
 `  N ) E. y  e.  ( EE `  N ) x  =/=  y )
 
Theoremaxlowdim2 23964* The lower two dimensional axiom. In any space where the dimension is greater than one, there are three non-colinear points. Axiom A8 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 15-Apr-2013.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) E. z  e.  ( EE `  N )  -.  ( x  Btwn  <. y ,  z >.  \/  y  Btwn  <. z ,  x >.  \/  z  Btwn  <. x ,  y >. ) )
 
Theoremaxlowdim 23965* The general lower dimensional axiom. Take a dimension  N greater than or equal to three. Then, there are three non-colinear points in  N dimensional space that are equidistant from  N  -  1 distinct points. Derived from remarks in "Tarski's System of Geometry", by Alfred Tarski and Steven Givant, Bull. Symbolic Logic Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.)
 |-  ( N  e.  ( ZZ>= `  3 )  ->  E. p E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) E. z  e.  ( EE `  N ) ( p :
 ( 1 ... ( N  -  1 ) )
 -1-1-> ( EE `  N )  /\  A. i  e.  ( 2 ... ( N  -  1 ) ) ( <. ( p `  1 ) ,  x >.Cgr
 <. ( p `  i
 ) ,  x >.  /\ 
 <. ( p `  1
 ) ,  y >.Cgr <.
 ( p `  i
 ) ,  y >.  /\ 
 <. ( p `  1
 ) ,  z >.Cgr <.
 ( p `  i
 ) ,  z >. ) 
 /\  -.  ( x  Btwn  <. y ,  z >.  \/  y  Btwn  <. z ,  x >.  \/  z  Btwn  <. x ,  y >. ) ) )
 
Theoremaxeuclidlem 23966* Lemma for axeuclid 23967. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)
 |-  (
 ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  T  e.  ( EE `  N ) ) )  /\  ( P  e.  ( 0 [,] 1 )  /\  Q  e.  ( 0 [,] 1
 )  /\  P  =/=  0 )  /\  A. i  e.  ( 1 ... N ) ( ( ( 1  -  P )  x.  ( A `  i ) )  +  ( P  x.  ( T `  i ) ) )  =  ( ( ( 1  -  Q )  x.  ( B `  i ) )  +  ( Q  x.  ( C `  i ) ) ) )  ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) E. r  e.  ( 0 [,] 1 ) E. s  e.  ( 0 [,] 1
 ) E. u  e.  ( 0 [,] 1
 ) A. i  e.  (
 1 ... N ) ( ( B `  i
 )  =  ( ( ( 1  -  r
 )  x.  ( A `
  i ) )  +  ( r  x.  ( x `  i
 ) ) )  /\  ( C `  i )  =  ( ( ( 1  -  s )  x.  ( A `  i ) )  +  ( s  x.  (
 y `  i )
 ) )  /\  ( T `  i )  =  ( ( ( 1  -  u )  x.  ( x `  i
 ) )  +  ( u  x.  ( y `  i ) ) ) ) )
 
Theoremaxeuclid 23967* Euclid's axiom. Take an angle  B A C and a point  D between  B and  C. Now, if you extend the segment  A D to a point  T, then  T lies between two points  x and  y that lie on the angle. Axiom A10 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 9-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  T  e.  ( EE `  N ) ) )  ->  (
 ( D  Btwn  <. A ,  T >.  /\  D  Btwn  <. B ,  C >.  /\  A  =/=  D ) 
 ->  E. x  e.  ( EE `  N ) E. y  e.  ( EE `  N ) ( B 
 Btwn  <. A ,  x >.  /\  C  Btwn  <. A ,  y >.  /\  T  Btwn  <. x ,  y >. ) ) )
 
Theoremaxcontlem1 23968* Lemma for axcont 23980. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( x `  i )  =  (
 ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i
 ) ) ) ) ) }   =>    |-  F  =  { <. y ,  s >.  |  ( y  e.  D  /\  ( s  e.  (
 0 [,)  +oo )  /\  A. j  e.  ( 1
 ... N ) ( y `  j )  =  ( ( ( 1  -  s )  x.  ( Z `  j ) )  +  ( s  x.  ( U `  j ) ) ) ) ) }
 
Theoremaxcontlem2 23969* Lemma for axcont 23980. The idea here is to set up a mapping  F that will allow us to transfer dedekind 23454 to two sets of points. Here, we set up  F and show its domain and range. (Contributed by Scott Fenton, 17-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) )  /\  Z  =/=  U )  ->  F : D -1-1-onto-> ( 0 [,)  +oo ) )
 
Theoremaxcontlem3 23970* Lemma for axcont 23980. Given the separation assumption,  B is a subset of  D. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   =>    |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N ) 
 /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/=  U ) )  ->  B  C_  D )
 
Theoremaxcontlem4 23971* Lemma for axcont 23980. Given the separation assumption,  A is a subset of  D. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   =>    |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N ) 
 /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  A  C_  D )
 
Theoremaxcontlem5 23972* Lemma for axcont 23980. Compute the value of  F. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  P  e.  D )  ->  ( ( F `
  P )  =  T  <->  ( T  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  (
 ( ( 1  -  T )  x.  ( Z `  i ) )  +  ( T  x.  ( U `  i ) ) ) ) ) )
 
Theoremaxcontlem6 23973* Lemma for axcont 23980. State the defining properties of the value of  F (Contributed by Scott Fenton, 19-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  P  e.  D )  ->  ( ( F `
  P )  e.  ( 0 [,)  +oo )  /\  A. i  e.  ( 1 ... N ) ( P `  i )  =  (
 ( ( 1  -  ( F `  P ) )  x.  ( Z `
  i ) )  +  ( ( F `
  P )  x.  ( U `  i
 ) ) ) ) )
 
Theoremaxcontlem7 23974* Lemma for axcont 23980. Given two points in  D, one preceeds the other iff its scaling constant is less than the other point's. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  ( P  e.  D  /\  Q  e.  D ) )  ->  ( P 
 Btwn  <. Z ,  Q >.  <-> 
 ( F `  P )  <_  ( F `  Q ) ) )
 
Theoremaxcontlem8 23975* Lemma for axcont 23980. A point in  D is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( ( N  e.  NN  /\  Z  e.  ( EE `  N )  /\  U  e.  ( EE `  N ) ) 
 /\  Z  =/=  U )  /\  ( P  e.  D  /\  Q  e.  D  /\  R  e.  D ) )  ->  ( (
 ( F `  P )  <_  ( F `  Q )  /\  ( F `
  Q )  <_  ( F `  R ) )  ->  Q  Btwn  <. P ,  R >. ) )
 
Theoremaxcontlem9 23976* Lemma for axcont 23980. Given the separation assumption, all values of  F over  A are less than or equal to all values of  F over  B. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  A. n  e.  ( F " A ) A. m  e.  ( F " B ) n  <_  m )
 
Theoremaxcontlem10 23977* Lemma for axcont 23980. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }   &    |-  F  =  { <. x ,  t >.  |  ( x  e.  D  /\  ( t  e.  (
 0 [,)  +oo )  /\  A. i  e.  ( 1
 ... N ) ( x `  i )  =  ( ( ( 1  -  t )  x.  ( Z `  i ) )  +  ( t  x.  ( U `  i ) ) ) ) ) }   =>    |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b  Btwn  <. x ,  y >. )
 
Theoremaxcontlem11 23978* Lemma for axcont 23980. Eliminate the hypotheses from axcontlem10 23977. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  (
 ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  B  =/=  (/) )  /\  Z  =/=  U ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b  Btwn  <. x ,  y >. )
 
Theoremaxcontlem12 23979* Lemma for axcont 23980. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  (
 ( ( N  e.  NN  /\  ( A  C_  ( EE `  N ) 
 /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  Z  e.  ( EE `  N ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b 
 Btwn  <. x ,  y >. )
 
Theoremaxcont 23980* The axiom of continuity. Take two sets of points  A and 
B. If all the points in  A come before the points of  B on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  E. a  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  x  Btwn  <. a ,  y >. ) )  ->  E. b  e.  ( EE `  N ) A. x  e.  A  A. y  e.  B  b 
 Btwn  <. x ,  y >. )
 
18.7.32  Congruence properties
 
Syntaxcofs 23981 Declare the syntax for the outer five segment configuration.
 class  OuterFiveSeg
 
Definitiondf-ofs 23982* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (ax5seg 23942). See brofs 24004 and 5segofs 24005 for how it is used. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.)
 |-  OuterFiveSeg  =  { <. p ,  q >.  |  E. n  e.  NN  E. a  e.  ( EE `  n ) E. b  e.  ( EE `  n ) E. c  e.  ( EE `  n ) E. d  e.  ( EE `  n ) E. x  e.  ( EE `  n ) E. y  e.  ( EE `  n ) E. z  e.  ( EE `  n ) E. w  e.  ( EE `  n ) ( p  =  <. <. a ,  b >. ,  <. c ,  d >. >.  /\  q  =  <.
 <. x ,  y >. , 
 <. z ,  w >. >.  /\  ( ( b  Btwn  <.
 a ,  c >.  /\  y  Btwn  <. x ,  z >. )  /\  ( <. a ,  b >.Cgr <. x ,  y >.  /\ 
 <. b ,  c >.Cgr <.
 y ,  z >. ) 
 /\  ( <. a ,  d >.Cgr <. x ,  w >.  /\  <. b ,  d >.Cgr
 <. y ,  w >. ) ) ) }
 
Theoremcgrrflx2d 23983 Deduction form of axcgrrflx 23918. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   =>    |-  ( ph  ->  <. A ,  B >.Cgr <. B ,  A >. )
 
Theoremcgrtr4d 23984 Deduction form of axcgrtr 23919. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. C ,  D >. )   &    |-  ( ph  ->  <. A ,  B >.Cgr <. E ,  F >. )   =>    |-  ( ph  ->  <. C ,  D >.Cgr <. E ,  F >. )
 
Theoremcgrtr4and 23985 Deduction form of axcgrtr 23919. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )
 
Theoremcgrrflx 23986 Reflexivity law for congruence. Theorem 2.1 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  A  e.  ( EE
 `  N )  /\  B  e.  ( EE `  N ) )  ->  <. A ,  B >.Cgr <. A ,  B >. )
 
Theoremcgrrflxd 23987 Deduction form of cgrrflx 23986. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   =>    |-  ( ph  ->  <. A ,  B >.Cgr <. A ,  B >. )
 
Theoremcgrcomim 23988 Congruence commutes on the two sides. Implication version. Theorem 2.2 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  ->  <. C ,  D >.Cgr <. A ,  B >. ) )
 
Theoremcgrcom 23989 Congruence commutes between the two sides. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. C ,  D >.Cgr <. A ,  B >. ) )
 
Theoremcgrcomand 23990 Deduction form of cgrcom 23989. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. A ,  B >. )
 
Theoremcgrtr 23991 Transitivity law for congruence. Theorem 2.3 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 24-Sep-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.Cgr <. C ,  D >.  /\ 
 <. C ,  D >.Cgr <. E ,  F >. ) 
 ->  <. A ,  B >.Cgr
 <. E ,  F >. ) )
 
Theoremcgrtrand 23992 Deduction form of cgrtr 23991. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )   &    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )
 
Theoremcgrtr3 23993 Transitivity law for congruence. (Contributed by Scott Fenton, 7-Oct-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) 
 /\  F  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.Cgr <. E ,  F >.  /\ 
 <. C ,  D >.Cgr <. E ,  F >. ) 
 ->  <. A ,  B >.Cgr
 <. C ,  D >. ) )
 
Theoremcgrtr3and 23994 Deduction form of cgrtr3 23993. (Contributed by Scott Fenton, 13-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ph  ->  E  e.  ( EE `  N ) )   &    |-  ( ph  ->  F  e.  ( EE `  N ) )   &    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. E ,  F >. )   &    |-  ( ( ph  /\  ps )  ->  <. C ,  D >.Cgr
 <. E ,  F >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. C ,  D >. )
 
Theoremcgrcoml 23995 Congruence commutes on the left. Biconditional version of Theorem 2.4 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. B ,  A >.Cgr <. C ,  D >. ) )
 
Theoremcgrcomr 23996 Congruence commutes on the right. Biconditional version of Theorem 2.5 of [Schwabhauser] p. 27. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. A ,  B >.Cgr <. D ,  C >. ) )
 
Theoremcgrcomlr 23997 Congruence commutes on both sides. (Contributed by Scott Fenton, 12-Jun-2013.)
 |-  (
 ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) ) 
 ->  ( <. A ,  B >.Cgr
 <. C ,  D >.  <->  <. B ,  A >.Cgr <. D ,  C >. ) )
 
Theoremcgrcomland 23998 Deduction form of cgrcoml 23995. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. B ,  A >.Cgr
 <. C ,  D >. )
 
Theoremcgrcomrand 23999 Deduction form of cgrcoml 23995. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. A ,  B >.Cgr
 <. D ,  C >. )
 
Theoremcgrcomlrand 24000 Deduction form of cgrcomlr 23997. (Contributed by Scott Fenton, 14-Oct-2013.)
 |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  A  e.  ( EE `  N ) )   &    |-  ( ph  ->  B  e.  ( EE `  N ) )   &    |-  ( ph  ->  C  e.  ( EE `  N ) )   &    |-  ( ph  ->  D  e.  ( EE `  N ) )   &    |-  ( ( ph  /\ 
 ps )  ->  <. A ,  B >.Cgr <. C ,  D >. )   =>    |-  ( ( ph  /\  ps )  ->  <. B ,  A >.Cgr
 <. D ,  C >. )
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