:: TRANSGEO semantic presentation

definition
let c1 be set ;
let c2, c3 be Permutation of c1;
redefine func * as c3 * c2 -> Permutation of a1;
coherence
c3 * c2 is Permutation of c1
proof end;
end;

theorem Th1: :: TRANSGEO:1
canceled;

theorem Th2: :: TRANSGEO:2
for b1 being non empty set
for b2 being Element of b1
for b3 being Permutation of b1 holds
ex b4 being Element of b1 st b3 . b4 = b2
proof end;

theorem Th3: :: TRANSGEO:3
canceled;

theorem Th4: :: TRANSGEO:4
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Permutation of b1 holds
( b4 . b2 = b3 iff (b4 " ) . b3 = b2 )
proof end;

definition
let c1 be non empty set ;
let c2, c3 be Permutation of c1;
func c2 \ c3 -> Permutation of a1 equals :: TRANSGEO:def 1
(a3 * a2) * (a3 " );
coherence
(c3 * c2) * (c3 " ) is Permutation of c1
;
end;

:: deftheorem Def1 defines \ TRANSGEO:def 1 :
for b1 being non empty set
for b2, b3 being Permutation of b1 holds b2 \ b3 = (b3 * b2) * (b3 " );

scheme :: TRANSGEO:sch 1
s1{ F1() -> non empty set , P1[ set , set ] } :
ex b1 being Permutation of F1() st
for b2, b3 being Element of F1() holds
( b1 . b2 = b3 iff P1[b2,b3] )
provided
E3: for b1 being Element of F1() holds
ex b2 being Element of F1() st P1[b1,b2] and E4: for b1 being Element of F1() holds
ex b2 being Element of F1() st P1[b2,b1] and E5: for b1, b2, b3 being Element of F1() holds
( P1[b1,b2] & P1[b3,b2] implies b1 = b3 ) and E6: for b1, b2, b3 being Element of F1() holds
( P1[b1,b2] & P1[b1,b3] implies b2 = b3 )
proof end;

theorem Th5: :: TRANSGEO:5
canceled;

theorem Th6: :: TRANSGEO:6
canceled;

theorem Th7: :: TRANSGEO:7
canceled;

theorem Th8: :: TRANSGEO:8
canceled;

theorem Th9: :: TRANSGEO:9
for b1 being non empty set
for b2 being Element of b1
for b3 being Permutation of b1 holds
( b3 . ((b3 " ) . b2) = b2 & (b3 " ) . (b3 . b2) = b2 ) by Th4;

theorem Th10: :: TRANSGEO:10
canceled;

theorem Th11: :: TRANSGEO:11
for b1 being non empty set
for b2 being Permutation of b1 holds b2 * (id b1) = (id b1) * b2
proof end;

Lemma3: for b1 being non empty set
for b2, b3, b4 being Permutation of b1 holds
( b2 * b3 = b2 * b4 implies b3 = b4 )
proof end;

Lemma4: for b1 being non empty set
for b2, b3, b4 being Permutation of b1 holds
( b2 * b3 = b4 * b3 implies b2 = b4 )
proof end;

theorem Th12: :: TRANSGEO:12
canceled;

theorem Th13: :: TRANSGEO:13
for b1 being non empty set
for b2, b3, b4 being Permutation of b1 holds
( ( b2 * b3 = b4 * b3 or b3 * b2 = b3 * b4 ) implies b2 = b4 ) by Lemma3, Lemma4;

theorem Th14: :: TRANSGEO:14
canceled;

theorem Th15: :: TRANSGEO:15
canceled;

theorem Th16: :: TRANSGEO:16
for b1 being non empty set
for b2, b3, b4 being Permutation of b1 holds (b2 * b3) \ b4 = (b2 \ b4) * (b3 \ b4)
proof end;

theorem Th17: :: TRANSGEO:17
for b1 being non empty set
for b2, b3 being Permutation of b1 holds (b2 " ) \ b3 = (b2 \ b3) "
proof end;

theorem Th18: :: TRANSGEO:18
for b1 being non empty set
for b2, b3, b4 being Permutation of b1 holds b2 \ (b3 * b4) = (b2 \ b4) \ b3
proof end;

theorem Th19: :: TRANSGEO:19
for b1 being non empty set
for b2 being Permutation of b1 holds (id b1) \ b2 = id b1
proof end;

theorem Th20: :: TRANSGEO:20
for b1 being non empty set
for b2 being Permutation of b1 holds b2 \ (id b1) = b2
proof end;

theorem Th21: :: TRANSGEO:21
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Permutation of b1 holds
( b3 . b2 = b2 implies (b3 \ b4) . (b4 . b2) = b4 . b2 )
proof end;

definition
let c1 be non empty set ;
let c2 be Permutation of c1;
let c3 be Relation of [:c1,c1:];
pred c2 is_FormalIz_of c3 means :Def2: :: TRANSGEO:def 2
for b1, b2 being Element of a1 holds [[b1,b2],[(a2 . b1),(a2 . b2)]] in a3;
end;

:: deftheorem Def2 defines is_FormalIz_of TRANSGEO:def 2 :
for b1 being non empty set
for b2 being Permutation of b1
for b3 being Relation of [:b1,b1:] holds
( b2 is_FormalIz_of b3 iff for b4, b5 being Element of b1 holds [[b4,b5],[(b2 . b4),(b2 . b5)]] in b3 );

theorem Th22: :: TRANSGEO:22
canceled;

theorem Th23: :: TRANSGEO:23
for b1 being non empty set
for b2 being Relation of [:b1,b1:] holds
( b2 is_reflexive_in [:b1,b1:] implies id b1 is_FormalIz_of b2 )
proof end;

theorem Th24: :: TRANSGEO:24
for b1 being non empty set
for b2 being Permutation of b1
for b3 being Relation of [:b1,b1:] holds
( b3 is_symmetric_in [:b1,b1:] & b2 is_FormalIz_of b3 implies b2 " is_FormalIz_of b3 )
proof end;

theorem Th25: :: TRANSGEO:25
for b1 being non empty set
for b2, b3 being Permutation of b1
for b4 being Relation of [:b1,b1:] holds
( b4 is_transitive_in [:b1,b1:] & b2 is_FormalIz_of b4 & b3 is_FormalIz_of b4 implies b2 * b3 is_FormalIz_of b4 )
proof end;

theorem Th26: :: TRANSGEO:26
for b1 being non empty set
for b2, b3 being Permutation of b1
for b4 being Relation of [:b1,b1:] holds
( ( for b5, b6, b7, b8, b9, b10 being Element of b1 holds
( [[b7,b8],[b5,b6]] in b4 & [[b5,b6],[b9,b10]] in b4 & b5 <> b6 implies [[b7,b8],[b9,b10]] in b4 ) ) & ( for b5, b6, b7 being Element of b1 holds [[b5,b5],[b6,b7]] in b4 ) & b2 is_FormalIz_of b4 & b3 is_FormalIz_of b4 implies b2 * b3 is_FormalIz_of b4 )
proof end;

definition
let c1 be non empty set ;
let c2 be Permutation of c1;
let c3 be Relation of [:c1,c1:];
pred c2 is_automorphism_of c3 means :Def3: :: TRANSGEO:def 3
for b1, b2, b3, b4 being Element of a1 holds
( [[b1,b2],[b3,b4]] in a3 iff [[(a2 . b1),(a2 . b2)],[(a2 . b3),(a2 . b4)]] in a3 );
end;

:: deftheorem Def3 defines is_automorphism_of TRANSGEO:def 3 :
for b1 being non empty set
for b2 being Permutation of b1
for b3 being Relation of [:b1,b1:] holds
( b2 is_automorphism_of b3 iff for b4, b5, b6, b7 being Element of b1 holds
( [[b4,b5],[b6,b7]] in b3 iff [[(b2 . b4),(b2 . b5)],[(b2 . b6),(b2 . b7)]] in b3 ) );

theorem Th27: :: TRANSGEO:27
canceled;

theorem Th28: :: TRANSGEO:28
for b1 being non empty set
for b2 being Relation of [:b1,b1:] holds id b1 is_automorphism_of b2
proof end;

theorem Th29: :: TRANSGEO:29
for b1 being non empty set
for b2 being Permutation of b1
for b3 being Relation of [:b1,b1:] holds
( b2 is_automorphism_of b3 implies b2 " is_automorphism_of b3 )
proof end;

theorem Th30: :: TRANSGEO:30
for b1 being non empty set
for b2, b3 being Permutation of b1
for b4 being Relation of [:b1,b1:] holds
( b2 is_automorphism_of b4 & b3 is_automorphism_of b4 implies b3 * b2 is_automorphism_of b4 )
proof end;

theorem Th31: :: TRANSGEO:31
for b1 being non empty set
for b2 being Permutation of b1
for b3 being Relation of [:b1,b1:] holds
( b3 is_symmetric_in [:b1,b1:] & b3 is_transitive_in [:b1,b1:] & b2 is_FormalIz_of b3 implies b2 is_automorphism_of b3 )
proof end;

theorem Th32: :: TRANSGEO:32
for b1 being non empty set
for b2 being Permutation of b1
for b3 being Relation of [:b1,b1:] holds
( ( for b4, b5, b6, b7, b8, b9 being Element of b1 holds
( [[b6,b7],[b4,b5]] in b3 & [[b4,b5],[b8,b9]] in b3 & b4 <> b5 implies [[b6,b7],[b8,b9]] in b3 ) ) & ( for b4, b5, b6 being Element of b1 holds [[b4,b4],[b5,b6]] in b3 ) & b3 is_symmetric_in [:b1,b1:] & b2 is_FormalIz_of b3 implies b2 is_automorphism_of b3 )
proof end;

theorem Th33: :: TRANSGEO:33
for b1 being non empty set
for b2, b3 being Permutation of b1
for b4 being Relation of [:b1,b1:] holds
( b2 is_FormalIz_of b4 & b3 is_automorphism_of b4 implies b2 \ b3 is_FormalIz_of b4 )
proof end;

definition
let c1 be non empty AffinStruct ;
let c2 be Permutation of the carrier of c1;
pred c2 is_DIL_of c1 means :Def4: :: TRANSGEO:def 4
a2 is_FormalIz_of the CONGR of a1;
end;

:: deftheorem Def4 defines is_DIL_of TRANSGEO:def 4 :
for b1 being non empty AffinStruct
for b2 being Permutation of the carrier of b1 holds
( b2 is_DIL_of b1 iff b2 is_FormalIz_of the CONGR of b1 );

theorem Th34: :: TRANSGEO:34
canceled;

theorem Th35: :: TRANSGEO:35
for b1 being non empty AffinStruct
for b2 being Permutation of the carrier of b1 holds
( b2 is_DIL_of b1 iff for b3, b4 being Element of b1 holds b3,b4 // b2 . b3,b2 . b4 )
proof end;

definition
let c1 be non empty AffinStruct ;
attr a1 is CongrSpace-like means :Def5: :: TRANSGEO:def 5
( ( for b1, b2, b3, b4, b5, b6 being Element of a1 holds
( b1,b2 // b5,b6 & b5,b6 // b3,b4 & b5 <> b6 implies b1,b2 // b3,b4 ) ) & ( for b1, b2, b3 being Element of a1 holds b1,b1 // b2,b3 ) & ( for b1, b2, b3, b4 being Element of a1 holds
( b1,b2 // b3,b4 implies b3,b4 // b1,b2 ) ) & ( for b1, b2 being Element of a1 holds b1,b2 // b1,b2 ) );
end;

:: deftheorem Def5 defines CongrSpace-like TRANSGEO:def 5 :
for b1 being non empty AffinStruct holds
( b1 is CongrSpace-like iff ( ( for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2,b3 // b6,b7 & b6,b7 // b4,b5 & b6 <> b7 implies b2,b3 // b4,b5 ) ) & ( for b2, b3, b4 being Element of b1 holds b2,b2 // b3,b4 ) & ( for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b4,b5 // b2,b3 ) ) & ( for b2, b3 being Element of b1 holds b2,b3 // b2,b3 ) ) );

registration
cluster non empty strict CongrSpace-like AffinStruct ;
existence
ex b1 being non empty AffinStruct st
( b1 is strict & b1 is CongrSpace-like )
proof end;
end;

definition
mode CongrSpace is non empty CongrSpace-like AffinStruct ;
end;

Lemma16: for b1 being CongrSpace holds the CONGR of b1 is_reflexive_in [:the carrier of b1,the carrier of b1:]
proof end;

Lemma17: for b1 being CongrSpace holds the CONGR of b1 is_symmetric_in [:the carrier of b1,the carrier of b1:]
proof end;

theorem Th36: :: TRANSGEO:36
canceled;

theorem Th37: :: TRANSGEO:37
for b1 being CongrSpace holds id the carrier of b1 is_DIL_of b1
proof end;

theorem Th38: :: TRANSGEO:38
for b1 being CongrSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_DIL_of b1 implies b2 " is_DIL_of b1 )
proof end;

theorem Th39: :: TRANSGEO:39
for b1 being CongrSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_DIL_of b1 & b3 is_DIL_of b1 implies b2 * b3 is_DIL_of b1 )
proof end;

theorem Th40: :: TRANSGEO:40
for b1 being OAffinSpace holds b1 is CongrSpace-like
proof end;

definition
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
attr a2 is positive_dilatation means :Def6: :: TRANSGEO:def 6
a2 is_DIL_of a1;
end;

:: deftheorem Def6 defines positive_dilatation TRANSGEO:def 6 :
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is positive_dilatation iff b2 is_DIL_of b1 );

notation
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
synonym c2 is_CDil for positive_dilatation c2;
end;

theorem Th41: :: TRANSGEO:41
canceled;

theorem Th42: :: TRANSGEO:42
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_CDil iff for b3, b4 being Element of b1 holds b3,b4 // b2 . b3,b2 . b4 )
proof end;

definition
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
attr a2 is negative_dilatation means :Def7: :: TRANSGEO:def 7
for b1, b2 being Element of a1 holds b1,b2 // a2 . b2,a2 . b1;
end;

:: deftheorem Def7 defines negative_dilatation TRANSGEO:def 7 :
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is negative_dilatation iff for b3, b4 being Element of b1 holds b3,b4 // b2 . b4,b2 . b3 );

notation
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
synonym c2 is_SDil for negative_dilatation c2;
end;

theorem Th43: :: TRANSGEO:43
canceled;

theorem Th44: :: TRANSGEO:44
for b1 being OAffinSpace holds id the carrier of b1 is_CDil
proof end;

theorem Th45: :: TRANSGEO:45
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_CDil implies b2 " is_CDil )
proof end;

theorem Th46: :: TRANSGEO:46
for b1 being OAffinSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_CDil & b3 is_CDil implies b2 * b3 is_CDil )
proof end;

theorem Th47: :: TRANSGEO:47
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
not ( b2 is_SDil & b2 is_CDil )
proof end;

theorem Th48: :: TRANSGEO:48
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_SDil implies b2 " is_SDil )
proof end;

theorem Th49: :: TRANSGEO:49
for b1 being OAffinSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_CDil & b3 is_SDil implies ( b2 * b3 is_SDil & b3 * b2 is_SDil ) )
proof end;

definition
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
attr a2 is dilatation means :Def8: :: TRANSGEO:def 8
a2 is_FormalIz_of lambda the CONGR of a1;
end;

:: deftheorem Def8 defines dilatation TRANSGEO:def 8 :
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is dilatation iff b2 is_FormalIz_of lambda the CONGR of b1 );

notation
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
synonym c2 is_Dil for dilatation c2;
end;

theorem Th50: :: TRANSGEO:50
canceled;

theorem Th51: :: TRANSGEO:51
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil iff for b3, b4 being Element of b1 holds b3,b4 '||' b2 . b3,b2 . b4 )
proof end;

theorem Th52: :: TRANSGEO:52
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( ( b2 is_CDil or b2 is_SDil ) implies b2 is_Dil )
proof end;

theorem Th53: :: TRANSGEO:53
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
not ( b2 is_Dil & ( for b3 being Permutation of the carrier of (Lambda b1) holds
not ( b2 = b3 & b3 is_DIL_of Lambda b1 ) ) )
proof end;

theorem Th54: :: TRANSGEO:54
for b1 being OAffinSpace
for b2 being Permutation of the carrier of (Lambda b1) holds
not ( b2 is_DIL_of Lambda b1 & ( for b3 being Permutation of the carrier of b1 holds
not ( b2 = b3 & b3 is_Dil ) ) )
proof end;

theorem Th55: :: TRANSGEO:55
for b1 being OAffinSpace holds id the carrier of b1 is_Dil
proof end;

theorem Th56: :: TRANSGEO:56
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies b2 " is_Dil )
proof end;

theorem Th57: :: TRANSGEO:57
for b1 being OAffinSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_Dil & b3 is_Dil implies b2 * b3 is_Dil )
proof end;

theorem Th58: :: TRANSGEO:58
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies for b3, b4, b5, b6 being Element of b1 holds
( b3,b4 '||' b5,b6 iff b2 . b3,b2 . b4 '||' b2 . b5,b2 . b6 ) )
proof end;

theorem Th59: :: TRANSGEO:59
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies for b3, b4, b5 being Element of b1 holds
( LIN b3,b4,b5 iff LIN b2 . b3,b2 . b4,b2 . b5 ) )
proof end;

theorem Th60: :: TRANSGEO:60
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Dil & LIN b2,b4 . b2,b3 implies LIN b2,b4 . b2,b4 . b3 )
proof end;

theorem Th61: :: TRANSGEO:61
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 '||' b4,b5 & not b2,b4 '||' b3,b5 & ( for b6 being Element of b1 holds
not ( LIN b2,b4,b6 & LIN b3,b5,b6 ) ) )
proof end;

theorem Th62: :: TRANSGEO:62
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies ( ( b2 = id the carrier of b1 or for b3 being Element of b1 holds
b2 . b3 <> b3 ) iff for b3, b4 being Element of b1 holds b3,b2 . b3 '||' b4,b2 . b4 ) )
proof end;

theorem Th63: :: TRANSGEO:63
for b1 being OAffinSpace
for b2, b3, b4 being Element of b1
for b5 being Permutation of the carrier of b1 holds
( b5 is_Dil & b5 . b2 = b2 & b5 . b3 = b3 & not LIN b2,b3,b4 implies b5 . b4 = b4 )
proof end;

theorem Th64: :: TRANSGEO:64
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Dil & b4 . b2 = b2 & b4 . b3 = b3 & b2 <> b3 implies b4 = id the carrier of b1 )
proof end;

theorem Th65: :: TRANSGEO:65
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4, b5 being Permutation of the carrier of b1 holds
not ( b4 is_Dil & b5 is_Dil & b4 . b2 = b5 . b2 & b4 . b3 = b5 . b3 & not b2 = b3 & not b4 = b5 )
proof end;

definition
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
attr a2 is translation means :Def9: :: TRANSGEO:def 9
( a2 is_Dil & ( a2 = id the carrier of a1 or for b1 being Element of a1 holds
b1 <> a2 . b1 ) );
end;

:: deftheorem Def9 defines translation TRANSGEO:def 9 :
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is translation iff ( b2 is_Dil & ( b2 = id the carrier of b1 or for b3 being Element of b1 holds
b3 <> b2 . b3 ) ) );

notation
let c1 be OAffinSpace;
let c2 be Permutation of the carrier of c1;
synonym c2 is_Tr for translation c2;
end;

theorem Th66: :: TRANSGEO:66
canceled;

theorem Th67: :: TRANSGEO:67
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies ( b2 is_Tr iff for b3, b4 being Element of b1 holds b3,b2 . b3 '||' b4,b2 . b4 ) )
proof end;

theorem Th68: :: TRANSGEO:68
canceled;

theorem Th69: :: TRANSGEO:69
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4, b5 being Permutation of the carrier of b1 holds
( b4 is_Tr & b5 is_Tr & b4 . b2 = b5 . b2 & not LIN b2,b4 . b2,b3 implies b4 . b3 = b5 . b3 )
proof end;

theorem Th70: :: TRANSGEO:70
for b1 being OAffinSpace
for b2 being Element of b1
for b3, b4 being Permutation of the carrier of b1 holds
( b3 is_Tr & b4 is_Tr & b3 . b2 = b4 . b2 implies b3 = b4 )
proof end;

theorem Th71: :: TRANSGEO:71
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Tr implies b2 " is_Tr )
proof end;

theorem Th72: :: TRANSGEO:72
for b1 being OAffinSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_Tr & b3 is_Tr implies b2 * b3 is_Tr )
proof end;

Lemma42: for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & not LIN b2,b4 . b2,b3 implies ( b2,b3 // b4 . b2,b4 . b3 & b2,b4 . b2 // b3,b4 . b3 ) )
proof end;

Lemma43: for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & b2 <> b4 . b2 & LIN b2,b4 . b2,b3 implies b2,b4 . b2 // b3,b4 . b3 )
proof end;

Lemma44: for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & Mid b2,b4 . b2,b3 & b2 <> b4 . b2 implies b2,b3 // b4 . b2,b4 . b3 )
proof end;

Lemma45: for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & b2 <> b4 . b2 & b3 <> b4 . b2 & Mid b2,b3,b4 . b2 implies b2,b3 // b4 . b2,b4 . b3 )
proof end;

Lemma46: for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Tr & b2 <> b4 . b2 & LIN b2,b4 . b2,b3 implies b2,b3 // b4 . b2,b4 . b3 )
proof end;

theorem Th73: :: TRANSGEO:73
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Tr implies b2 is_CDil )
proof end;

theorem Th74: :: TRANSGEO:74
for b1 being OAffinSpace
for b2, b3, b4 being Element of b1
for b5 being Permutation of the carrier of b1 holds
( b5 is_Dil & b5 . b2 = b2 & Mid b3,b2,b5 . b3 & not LIN b2,b3,b4 implies Mid b4,b2,b5 . b4 )
proof end;

theorem Th75: :: TRANSGEO:75
for b1 being OAffinSpace
for b2, b3, b4 being Element of b1
for b5 being Permutation of the carrier of b1 holds
( b5 is_Dil & b5 . b2 = b2 & Mid b3,b2,b5 . b3 & b3 <> b2 implies Mid b4,b2,b5 . b4 )
proof end;

theorem Th76: :: TRANSGEO:76
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1
for b6 being Permutation of the carrier of b1 holds
( b6 is_Dil & b6 . b2 = b2 & b3 <> b2 & Mid b3,b2,b6 . b3 & not LIN b2,b4,b5 implies b4,b5 // b6 . b5,b6 . b4 )
proof end;

theorem Th77: :: TRANSGEO:77
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1
for b6 being Permutation of the carrier of b1 holds
( b6 is_Dil & b6 . b2 = b2 & b3 <> b2 & Mid b3,b2,b6 . b3 & LIN b2,b4,b5 implies b4,b5 // b6 . b5,b6 . b4 )
proof end;

theorem Th78: :: TRANSGEO:78
for b1 being OAffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Dil & b4 . b2 = b2 & b3 <> b2 & Mid b3,b2,b4 . b3 implies b4 is_SDil )
proof end;

theorem Th79: :: TRANSGEO:79
for b1 being OAffinSpace
for b2 being Element of b1
for b3 being Permutation of the carrier of b1 holds
( b3 is_Dil & b3 . b2 = b2 & ( for b4 being Element of b1 holds b2,b4 // b2,b3 . b4 ) implies for b4, b5 being Element of b1 holds b4,b5 // b3 . b4,b3 . b5 )
proof end;

theorem Th80: :: TRANSGEO:80
for b1 being OAffinSpace
for b2 being Permutation of the carrier of b1 holds
not ( b2 is_Dil & not b2 is_CDil & not b2 is_SDil )
proof end;

theorem Th81: :: TRANSGEO:81
canceled;

theorem Th82: :: TRANSGEO:82
for b1 being AffinSpace holds b1 is CongrSpace-like
proof end;

theorem Th83: :: TRANSGEO:83
for b1 being OAffinSpace holds
Lambda b1 is CongrSpace
proof end;

definition
let c1 be AffinSpace;
let c2 be Permutation of the carrier of c1;
attr a2 is dilatation means :Def10: :: TRANSGEO:def 10
a2 is_DIL_of a1;
end;

:: deftheorem Def10 defines dilatation TRANSGEO:def 10 :
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is dilatation iff b2 is_DIL_of b1 );

notation
let c1 be AffinSpace;
let c2 be Permutation of the carrier of c1;
synonym c2 is_Dil for dilatation c2;
end;

theorem Th84: :: TRANSGEO:84
canceled;

theorem Th85: :: TRANSGEO:85
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil iff for b3, b4 being Element of b1 holds b3,b4 // b2 . b3,b2 . b4 )
proof end;

theorem Th86: :: TRANSGEO:86
for b1 being AffinSpace holds id the carrier of b1 is_Dil
proof end;

theorem Th87: :: TRANSGEO:87
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies b2 " is_Dil )
proof end;

theorem Th88: :: TRANSGEO:88
for b1 being AffinSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_Dil & b3 is_Dil implies b2 * b3 is_Dil )
proof end;

theorem Th89: :: TRANSGEO:89
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies for b3, b4, b5, b6 being Element of b1 holds
( b3,b4 // b5,b6 iff b2 . b3,b2 . b4 // b2 . b5,b2 . b6 ) )
proof end;

theorem Th90: :: TRANSGEO:90
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies for b3, b4, b5 being Element of b1 holds
( LIN b3,b4,b5 iff LIN b2 . b3,b2 . b4,b2 . b5 ) )
proof end;

theorem Th91: :: TRANSGEO:91
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Dil & LIN b2,b4 . b2,b3 implies LIN b2,b4 . b2,b4 . b3 )
proof end;

theorem Th92: :: TRANSGEO:92
for b1 being AffinSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( b2,b3 // b4,b5 & not b2,b4 // b3,b5 & ( for b6 being Element of b1 holds
not ( LIN b2,b4,b6 & LIN b3,b5,b6 ) ) )
proof end;

theorem Th93: :: TRANSGEO:93
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies ( ( b2 = id the carrier of b1 or for b3 being Element of b1 holds
b2 . b3 <> b3 ) iff for b3, b4 being Element of b1 holds b3,b2 . b3 // b4,b2 . b4 ) )
proof end;

theorem Th94: :: TRANSGEO:94
for b1 being AffinSpace
for b2, b3, b4 being Element of b1
for b5 being Permutation of the carrier of b1 holds
( b5 is_Dil & b5 . b2 = b2 & b5 . b3 = b3 & not LIN b2,b3,b4 implies b5 . b4 = b4 )
proof end;

theorem Th95: :: TRANSGEO:95
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Dil & b4 . b2 = b2 & b4 . b3 = b3 & b2 <> b3 implies b4 = id the carrier of b1 )
proof end;

theorem Th96: :: TRANSGEO:96
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4, b5 being Permutation of the carrier of b1 holds
not ( b4 is_Dil & b5 is_Dil & b4 . b2 = b5 . b2 & b4 . b3 = b5 . b3 & not b2 = b3 & not b4 = b5 )
proof end;

theorem Th97: :: TRANSGEO:97
for b1 being AffinSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not LIN b2,b3,b4 & b2,b3 // b4,b5 & b2,b3 // b4,b6 & b2,b4 // b3,b5 & b2,b4 // b3,b6 implies b5 = b6 )
proof end;

definition
let c1 be AffinSpace;
let c2 be Permutation of the carrier of c1;
attr a2 is translation means :Def11: :: TRANSGEO:def 11
( a2 is_Dil & ( a2 = id the carrier of a1 or for b1 being Element of a1 holds
b1 <> a2 . b1 ) );
end;

:: deftheorem Def11 defines translation TRANSGEO:def 11 :
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is translation iff ( b2 is_Dil & ( b2 = id the carrier of b1 or for b3 being Element of b1 holds
b3 <> b2 . b3 ) ) );

notation
let c1 be AffinSpace;
let c2 be Permutation of the carrier of c1;
synonym c2 is_Tr for translation c2;
end;

theorem Th98: :: TRANSGEO:98
canceled;

theorem Th99: :: TRANSGEO:99
for b1 being AffinSpace holds id the carrier of b1 is_Tr
proof end;

theorem Th100: :: TRANSGEO:100
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Dil implies ( b2 is_Tr iff for b3, b4 being Element of b1 holds b3,b2 . b3 // b4,b2 . b4 ) )
proof end;

theorem Th101: :: TRANSGEO:101
canceled;

theorem Th102: :: TRANSGEO:102
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4, b5 being Permutation of the carrier of b1 holds
( b4 is_Tr & b5 is_Tr & b4 . b2 = b5 . b2 & not LIN b2,b4 . b2,b3 implies b4 . b3 = b5 . b3 )
proof end;

theorem Th103: :: TRANSGEO:103
for b1 being AffinSpace
for b2 being Element of b1
for b3, b4 being Permutation of the carrier of b1 holds
( b3 is_Tr & b4 is_Tr & b3 . b2 = b4 . b2 implies b3 = b4 )
proof end;

theorem Th104: :: TRANSGEO:104
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Tr implies b2 " is_Tr )
proof end;

theorem Th105: :: TRANSGEO:105
for b1 being AffinSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_Tr & b3 is_Tr implies b2 * b3 is_Tr )
proof end;

definition
let c1 be AffinSpace;
let c2 be Permutation of the carrier of c1;
attr a2 is collineation means :: TRANSGEO:def 12
a2 is_automorphism_of the CONGR of a1;
end;

:: deftheorem Def12 defines collineation TRANSGEO:def 12 :
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is collineation iff b2 is_automorphism_of the CONGR of b1 );

notation
let c1 be AffinSpace;
let c2 be Permutation of the carrier of c1;
synonym c2 is_Col for collineation c2;
end;

theorem Th106: :: TRANSGEO:106
canceled;

theorem Th107: :: TRANSGEO:107
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1 holds
( b2 is_Col iff for b3, b4, b5, b6 being Element of b1 holds
( b3,b4 // b5,b6 iff b2 . b3,b2 . b4 // b2 . b5,b2 . b6 ) )
proof end;

theorem Th108: :: TRANSGEO:108
for b1 being AffinSpace
for b2, b3, b4 being Element of b1
for b5 being Permutation of the carrier of b1 holds
( b5 is_Col implies ( LIN b2,b3,b4 iff LIN b5 . b2,b5 . b3,b5 . b4 ) )
proof end;

theorem Th109: :: TRANSGEO:109
for b1 being AffinSpace
for b2, b3 being Permutation of the carrier of b1 holds
( b2 is_Col & b3 is_Col implies ( b2 " is_Col & b2 * b3 is_Col & id the carrier of b1 is_Col ) )
proof end;

theorem Th110: :: TRANSGEO:110
for b1 being AffinSpace
for b2 being Element of b1
for b3 being Permutation of the carrier of b1
for b4 being Subset of b1 holds
( b2 in b4 implies b3 . b2 in b3 .: b4 )
proof end;

theorem Th111: :: TRANSGEO:111
for b1 being AffinSpace
for b2 being Element of b1
for b3 being Permutation of the carrier of b1
for b4 being Subset of b1 holds
( b2 in b3 .: b4 iff ex b5 being Element of b1 st
( b5 in b4 & b3 . b5 = b2 ) )
proof end;

theorem Th112: :: TRANSGEO:112
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1
for b3, b4 being Subset of b1 holds
( b2 .: b3 = b2 .: b4 implies b3 = b4 )
proof end;

theorem Th113: :: TRANSGEO:113
for b1 being AffinSpace
for b2, b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Col implies b4 .: (Line b2,b3) = Line (b4 . b2),(b4 . b3) )
proof end;

theorem Th114: :: TRANSGEO:114
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1
for b3 being Subset of b1 holds
( b2 is_Col & b3 is_line implies b2 .: b3 is_line )
proof end;

theorem Th115: :: TRANSGEO:115
for b1 being AffinSpace
for b2 being Permutation of the carrier of b1
for b3, b4 being Subset of b1 holds
( b2 is_Col & b3 // b4 implies b2 .: b3 // b2 .: b4 )
proof end;

theorem Th116: :: TRANSGEO:116
for b1 being AffinPlane
for b2 being Permutation of the carrier of b1 holds
( ( for b3 being Subset of b1 holds
( b3 is_line implies b2 .: b3 is_line ) ) implies b2 is_Col )
proof end;

theorem Th117: :: TRANSGEO:117
for b1 being AffinPlane
for b2 being Subset of b1
for b3 being Element of b1
for b4 being Permutation of the carrier of b1 holds
( b4 is_Col & b2 is_line & ( for b5 being Element of b1 holds
( b5 in b2 implies b4 . b5 = b5 ) ) & not b3 in b2 & b4 . b3 = b3 implies b4 = id the carrier of b1 )
proof end;