:: SEMI_AF1 semantic presentation
definition
let c
1 be non
empty AffinStruct ;
attr a
1 is
Semi_Affine_Space-like means :
Def1:
:: SEMI_AF1:def 1
( ( for b
1, b
2 being
Element of a
1 holds b
1,b
2 // b
2,b
1 ) & ( for b
1, b
2, b
3 being
Element of a
1 holds b
1,b
2 // b
3,b
3 ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1 <> b
2 & b
1,b
2 // b
3,b
4 & b
1,b
2 // b
5,b
6 implies b
3,b
4 // b
5,b
6 ) ) & ( for b
1, b
2, b
3 being
Element of a
1 holds
( b
1,b
2 // b
1,b
3 implies b
2,b
1 // b
2,b
3 ) ) & not for b
1, b
2, b
3 being
Element of a
1 holds b
1,b
2 // b
1,b
3 & ( for b
1, b
2, b
3 being
Element of a
1 holds
ex b
4 being
Element of a
1 st
( b
1,b
2 // b
3,b
4 & b
1,b
3 // b
2,b
4 ) ) & ( for b
1, b
2 being
Element of a
1 holds
ex b
3 being
Element of a
1 st
for b
4, b
5 being
Element of a
1 holds
( b
1,b
2 // b
1,b
3 & not for b
6 being
Element of a
1 holds
( b
1,b
3 // b
1,b
4 & not ( b
1,b
5 // b
1,b
6 & b
3,b
5 // b
4,b
6 ) ) ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1 holds
( not b
1,b
2 // b
1,b
4 & not b
1,b
2 // b
1,b
6 & b
1,b
2 // b
1,b
3 & b
1,b
4 // b
1,b
5 & b
1,b
6 // b
1,b
7 & b
2,b
4 // b
3,b
5 & b
2,b
6 // b
3,b
7 implies b
4,b
6 // b
5,b
7 ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( not b
1,b
2 // b
1,b
3 & not b
1,b
2 // b
1,b
5 & b
1,b
2 // b
3,b
4 & b
1,b
2 // b
5,b
6 & b
1,b
3 // b
2,b
4 & b
1,b
5 // b
2,b
6 implies b
3,b
5 // b
4,b
6 ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
1,b
3 & b
4,b
5 // b
4,b
6 & b
1,b
5 // b
2,b
4 & b
2,b
6 // b
3,b
5 implies b
3,b
4 // b
1,b
6 ) ) & ( for b
1, b
2, b
3, b
4 being
Element of a
1 holds
not ( not b
1,b
2 // b
1,b
3 & b
1,b
2 // b
3,b
4 & b
1,b
3 // b
2,b
4 & b
1,b
4 // b
2,b
3 ) ) );
end;
:: deftheorem Def1 defines Semi_Affine_Space-like SEMI_AF1:def 1 :
for b
1 being non
empty AffinStruct holds
( b
1 is
Semi_Affine_Space-like iff ( ( for b
2, b
3 being
Element of b
1 holds b
2,b
3 // b
3,b
2 ) & ( for b
2, b
3, b
4 being
Element of b
1 holds b
2,b
3 // b
4,b
4 ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & b
2,b
3 // b
4,b
5 & b
2,b
3 // b
6,b
7 implies b
4,b
5 // b
6,b
7 ) ) & ( for b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 implies b
3,b
2 // b
3,b
4 ) ) & not for b
2, b
3, b
4 being
Element of b
1 holds b
2,b
3 // b
2,b
4 & ( for b
2, b
3, b
4 being
Element of b
1 holds
ex b
5 being
Element of b
1 st
( b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 ) ) & ( for b
2, b
3 being
Element of b
1 holds
ex b
4 being
Element of b
1 st
for b
5, b
6 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 & not for b
7 being
Element of b
1 holds
( b
2,b
4 // b
2,b
5 & not ( b
2,b
6 // b
2,b
7 & b
4,b
6 // b
5,b
7 ) ) ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
5 & not b
2,b
3 // b
2,b
7 & b
2,b
3 // b
2,b
4 & b
2,b
5 // b
2,b
6 & b
2,b
7 // b
2,b
8 & b
3,b
5 // b
4,b
6 & b
3,b
7 // b
4,b
8 implies b
5,b
7 // b
6,b
8 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
4 & not b
2,b
3 // b
2,b
6 & b
2,b
3 // b
4,b
5 & b
2,b
3 // b
6,b
7 & b
2,b
4 // b
3,b
5 & b
2,b
6 // b
3,b
7 implies b
4,b
6 // b
5,b
7 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 & b
5,b
6 // b
5,b
7 & b
2,b
6 // b
3,b
5 & b
3,b
7 // b
4,b
6 implies b
4,b
5 // b
2,b
7 ) ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( not b
2,b
3 // b
2,b
4 & b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 & b
2,b
5 // b
3,b
4 ) ) ) );
theorem Th1: :: SEMI_AF1:1
canceled;
theorem Th2: :: SEMI_AF1:2
canceled;
theorem Th3: :: SEMI_AF1:3
canceled;
theorem Th4: :: SEMI_AF1:4
canceled;
theorem Th5: :: SEMI_AF1:5
canceled;
theorem Th6: :: SEMI_AF1:6
canceled;
theorem Th7: :: SEMI_AF1:7
canceled;
theorem Th8: :: SEMI_AF1:8
canceled;
theorem Th9: :: SEMI_AF1:9
canceled;
theorem Th10: :: SEMI_AF1:10
canceled;
theorem Th11: :: SEMI_AF1:11
canceled;
theorem Th12: :: SEMI_AF1:12
theorem Th13: :: SEMI_AF1:13
theorem Th14: :: SEMI_AF1:14
theorem Th15: :: SEMI_AF1:15
theorem Th16: :: SEMI_AF1:16
theorem Th17: :: SEMI_AF1:17
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies ( b
3,b
2 // b
4,b
5 & b
2,b
3 // b
5,b
4 & b
3,b
2 // b
5,b
4 & b
4,b
5 // b
2,b
3 & b
5,b
4 // b
2,b
3 & b
4,b
5 // b
3,b
2 & b
5,b
4 // b
3,b
2 ) )
theorem Th18: :: SEMI_AF1:18
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 implies ( b
2,b
4 // b
2,b
3 & b
3,b
2 // b
2,b
4 & b
2,b
3 // b
4,b
2 & b
2,b
4 // b
3,b
2 & b
3,b
2 // b
4,b
2 & b
4,b
2 // b
2,b
3 & b
4,b
2 // b
3,b
2 & b
3,b
2 // b
3,b
4 & b
2,b
3 // b
3,b
4 & b
3,b
2 // b
4,b
3 & b
3,b
4 // b
3,b
2 & b
2,b
3 // b
4,b
3 & b
4,b
3 // b
3,b
2 & b
3,b
4 // b
2,b
3 & b
4,b
3 // b
2,b
3 & b
4,b
2 // b
4,b
3 & b
2,b
4 // b
4,b
3 & b
4,b
2 // b
3,b
4 & b
2,b
4 // b
3,b
4 & b
4,b
3 // b
4,b
2 & b
3,b
4 // b
4,b
2 & b
4,b
3 // b
2,b
4 & b
3,b
4 // b
2,b
4 ) )
theorem Th19: :: SEMI_AF1:19
canceled;
theorem Th20: :: SEMI_AF1:20
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & b
4,b
5 // b
2,b
3 & b
2,b
3 // b
6,b
7 implies b
4,b
5 // b
6,b
7 )
theorem Th21: :: SEMI_AF1:21
theorem Th22: :: SEMI_AF1:22
theorem Th23: :: SEMI_AF1:23
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 & b
3,b
5 // b
3,b
4 & b
5,b
2 // b
5,b
4 implies b
2,b
3 // b
2,b
5 )
theorem Th24: :: SEMI_AF1:24
canceled;
theorem Th25: :: SEMI_AF1:25
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( not b
2,b
3 // b
2,b
4 & b
5 <> b
6 & b
5,b
6 // b
5,b
2 & b
5,b
6 // b
5,b
3 & b
5,b
6 // b
5,b
4 )
theorem Th26: :: SEMI_AF1:26
theorem Th27: :: SEMI_AF1:27
canceled;
theorem Th28: :: SEMI_AF1:28
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
4 implies ( not b
2,b
3 // b
4,b
2 & not b
3,b
2 // b
2,b
4 & not b
3,b
2 // b
4,b
2 & not b
2,b
4 // b
2,b
3 & not b
2,b
4 // b
3,b
2 & not b
4,b
2 // b
2,b
3 & not b
4,b
2 // b
3,b
2 & not b
3,b
2 // b
3,b
4 & not b
3,b
2 // b
4,b
3 & not b
2,b
3 // b
3,b
4 & not b
2,b
3 // b
4,b
3 & not b
3,b
4 // b
3,b
2 & not b
3,b
4 // b
2,b
3 & not b
4,b
3 // b
2,b
3 & not b
4,b
3 // b
3,b
2 & not b
4,b
3 // b
4,b
2 & not b
4,b
3 // b
2,b
4 & not b
3,b
4 // b
4,b
2 & not b
3,b
4 // b
2,b
4 & not b
4,b
2 // b
4,b
3 & not b
4,b
2 // b
3,b
4 & not b
2,b
4 // b
3,b
4 & not b
2,b
4 // b
4,b
3 ) )
theorem Th29: :: SEMI_AF1:29
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
not ( not b
2,b
3 // b
4,b
5 & b
2,b
3 // b
6,b
7 & b
4,b
5 // b
8,b
9 & b
6 <> b
7 & b
8 <> b
9 & b
6,b
7 // b
8,b
9 )
theorem Th30: :: SEMI_AF1:30
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
not ( not b
2,b
3 // b
2,b
4 & b
2,b
3 // b
5,b
6 & b
2,b
4 // b
5,b
7 & b
3,b
4 // b
6,b
7 & b
5 <> b
6 & b
5,b
6 // b
5,b
7 )
theorem Th31: :: SEMI_AF1:31
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
4 & b
2,b
4 // b
5,b
6 & b
3,b
4 // b
5,b
6 implies b
5 = b
6 )
theorem Th32: :: SEMI_AF1:32
theorem Th33: :: SEMI_AF1:33
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( not b
2,b
3 // b
2,b
4 & b
2,b
3 // b
5,b
6 & b
2,b
4 // b
5,b
7 & b
2,b
4 // b
5,b
8 & b
3,b
4 // b
6,b
7 & b
3,b
4 // b
6,b
8 implies b
7 = b
8 )
theorem Th34: :: SEMI_AF1:34
theorem Th35: :: SEMI_AF1:35
:: deftheorem Def2 defines is_collinear SEMI_AF1:def 2 :
theorem Th36: :: SEMI_AF1:36
canceled;
theorem Th37: :: SEMI_AF1:37
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3,b
4 is_collinear implies ( b
2,b
4,b
3 is_collinear & b
3,b
2,b
4 is_collinear & b
3,b
4,b
2 is_collinear & b
4,b
2,b
3 is_collinear & b
4,b
3,b
2 is_collinear ) )
theorem Th38: :: SEMI_AF1:38
canceled;
theorem Th39: :: SEMI_AF1:39
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
not ( not b
2,b
3,b
4 is_collinear & b
2,b
3 // b
5,b
6 & b
2,b
4 // b
5,b
7 & b
5 <> b
6 & b
5 <> b
7 & b
5,b
6,b
7 is_collinear )
theorem Th40: :: SEMI_AF1:40
theorem Th41: :: SEMI_AF1:41
theorem Th42: :: SEMI_AF1:42
theorem Th43: :: SEMI_AF1:43
theorem Th44: :: SEMI_AF1:44
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( not b
2,b
3,b
4 is_collinear & b
2,b
3 // b
4,b
5 & b
4 <> b
5 & b
4,b
5,b
6 is_collinear & b
2,b
3,b
6 is_collinear )
theorem Th45: :: SEMI_AF1:45
theorem Th46: :: SEMI_AF1:46
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 & b
2 <> b
4 & b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
4,b
6 is_collinear implies b
3,b
4 // b
5,b
6 )
theorem Th47: :: SEMI_AF1:47
canceled;
theorem Th48: :: SEMI_AF1:48
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
3 // b
4,b
5 & b
2,b
3,b
6 is_collinear & b
2,b
3,b
7 is_collinear & b
4,b
5,b
6 is_collinear & b
4,b
5,b
7 is_collinear implies b
6 = b
7 )
theorem Th49: :: SEMI_AF1:49
theorem Th50: :: SEMI_AF1:50
theorem Th51: :: SEMI_AF1:51
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
4,b
6 is_collinear & b
2,b
4,b
7 is_collinear & b
3,b
4 // b
5,b
6 & b
3,b
4 // b
5,b
6 & b
3,b
4 // b
5,b
7 implies b
6 = b
7 )
theorem Th52: :: SEMI_AF1:52
definition
let c
1 be
Semi_Affine_Space;
let c
2, c
3, c
4, c
5 be
Element of c
1;
pred parallelogram c
2,c
3,c
4,c
5 means :
Def3:
:: SEMI_AF1:def 3
( not a
2,a
3,a
4 is_collinear & a
2,a
3 // a
4,a
5 & a
2,a
4 // a
3,a
5 );
end;
:: deftheorem Def3 defines parallelogram SEMI_AF1:def 3 :
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
parallelogram b
2,b
3,b
4,b
5 iff ( not b
2,b
3,b
4 is_collinear & b
2,b
3 // b
4,b
5 & b
2,b
4 // b
3,b
5 ) );
theorem Th53: :: SEMI_AF1:53
canceled;
theorem Th54: :: SEMI_AF1:54
theorem Th55: :: SEMI_AF1:55
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
parallelogram b
2,b
3,b
4,b
5 implies ( not b
2,b
3,b
4 is_collinear & not b
3,b
2,b
5 is_collinear & not b
4,b
5,b
2 is_collinear & not b
5,b
4,b
3 is_collinear ) )
theorem Th56: :: SEMI_AF1:56
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
parallelogram b
2,b
3,b
4,b
5 implies ( not b
2,b
3,b
4 is_collinear & not b
2,b
4,b
3 is_collinear & not b
2,b
3,b
5 is_collinear & not b
2,b
5,b
3 is_collinear & not b
2,b
4,b
5 is_collinear & not b
2,b
5,b
4 is_collinear & not b
3,b
2,b
4 is_collinear & not b
3,b
4,b
2 is_collinear & not b
3,b
2,b
5 is_collinear & not b
3,b
5,b
2 is_collinear & not b
3,b
4,b
5 is_collinear & not b
3,b
5,b
4 is_collinear & not b
4,b
2,b
3 is_collinear & not b
4,b
3,b
2 is_collinear & not b
4,b
2,b
5 is_collinear & not b
4,b
5,b
2 is_collinear & not b
4,b
3,b
5 is_collinear & not b
4,b
5,b
3 is_collinear & not b
5,b
2,b
3 is_collinear & not b
5,b
3,b
2 is_collinear & not b
5,b
2,b
4 is_collinear & not b
5,b
4,b
2 is_collinear & not b
5,b
3,b
4 is_collinear & not b
5,b
4,b
3 is_collinear ) )
theorem Th57: :: SEMI_AF1:57
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not (
parallelogram b
2,b
3,b
4,b
5 & b
2,b
3,b
6 is_collinear & b
4,b
5,b
6 is_collinear )
theorem Th58: :: SEMI_AF1:58
theorem Th59: :: SEMI_AF1:59
theorem Th60: :: SEMI_AF1:60
theorem Th61: :: SEMI_AF1:61
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
parallelogram b
2,b
3,b
4,b
5 implies (
parallelogram b
2,b
4,b
3,b
5 &
parallelogram b
4,b
5,b
2,b
3 &
parallelogram b
3,b
2,b
5,b
4 &
parallelogram b
4,b
2,b
5,b
3 &
parallelogram b
5,b
3,b
4,b
2 &
parallelogram b
3,b
5,b
2,b
4 ) )
theorem Th62: :: SEMI_AF1:62
theorem Th63: :: SEMI_AF1:63
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
parallelogram b
2,b
3,b
4,b
5 &
parallelogram b
2,b
3,b
4,b
6 implies b
5 = b
6 )
theorem Th64: :: SEMI_AF1:64
theorem Th65: :: SEMI_AF1:65
theorem Th66: :: SEMI_AF1:66
theorem Th67: :: SEMI_AF1:67
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
parallelogram b
2,b
3,b
4,b
5 &
parallelogram b
2,b
3,b
6,b
7 implies b
4,b
6 // b
5,b
7 )
theorem Th68: :: SEMI_AF1:68
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
3,b
4 is_collinear &
parallelogram b
5,b
6,b
2,b
3 &
parallelogram b
5,b
6,b
4,b
7 implies
parallelogram b
2,b
3,b
4,b
7 )
theorem Th69: :: SEMI_AF1:69
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3,b
4 is_collinear & b
3 <> b
4 &
parallelogram b
2,b
5,b
3,b
6 &
parallelogram b
2,b
5,b
4,b
7 implies
parallelogram b
3,b
6,b
4,b
7 )
theorem Th70: :: SEMI_AF1:70
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
(
parallelogram b
2,b
3,b
4,b
5 &
parallelogram b
2,b
3,b
6,b
7 &
parallelogram b
4,b
5,b
8,b
9 implies b
6,b
8 // b
7,b
9 )
Lemma44:
for b1 being Semi_Affine_Space
for b2, b3 being Element of b1 holds
not ( b2 <> b3 & ( for b4, b5 being Element of b1 holds
not parallelogram b2,b3,b4,b5 ) )
theorem Th71: :: SEMI_AF1:71
definition
let c
1 be
Semi_Affine_Space;
let c
2, c
3, c
4, c
5 be
Element of c
1;
pred congr c
2,c
3,c
4,c
5 means :
Def4:
:: SEMI_AF1:def 4
not ( not ( a
2 = a
3 & a
4 = a
5 ) & ( for b
1, b
2 being
Element of a
1 holds
not (
parallelogram b
1,b
2,a
2,a
3 &
parallelogram b
1,b
2,a
4,a
5 ) ) );
end;
:: deftheorem Def4 defines congr SEMI_AF1:def 4 :
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 iff not ( not ( b
2 = b
3 & b
4 = b
5 ) & ( for b
6, b
7 being
Element of b
1 holds
not (
parallelogram b
6,b
7,b
2,b
3 &
parallelogram b
6,b
7,b
4,b
5 ) ) ) );
theorem Th72: :: SEMI_AF1:72
canceled;
theorem Th73: :: SEMI_AF1:73
theorem Th74: :: SEMI_AF1:74
theorem Th75: :: SEMI_AF1:75
theorem Th76: :: SEMI_AF1:76
theorem Th77: :: SEMI_AF1:77
theorem Th78: :: SEMI_AF1:78
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 & not b
2,b
3,b
4 is_collinear implies
parallelogram b
2,b
3,b
4,b
5 )
theorem Th79: :: SEMI_AF1:79
theorem Th80: :: SEMI_AF1:80
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 & b
2,b
3,b
4 is_collinear &
parallelogram b
6,b
7,b
2,b
3 implies
parallelogram b
6,b
7,b
4,b
5 )
theorem Th81: :: SEMI_AF1:81
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 &
congr b
2,b
3,b
4,b
6 implies b
5 = b
6 )
theorem Th82: :: SEMI_AF1:82
theorem Th83: :: SEMI_AF1:83
canceled;
theorem Th84: :: SEMI_AF1:84
theorem Th85: :: SEMI_AF1:85
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 &
congr b
2,b
3,b
6,b
7 implies
congr b
4,b
5,b
6,b
7 )
theorem Th86: :: SEMI_AF1:86
theorem Th87: :: SEMI_AF1:87
theorem Th88: :: SEMI_AF1:88
theorem Th89: :: SEMI_AF1:89
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 implies (
congr b
4,b
5,b
2,b
3 &
congr b
3,b
2,b
5,b
4 &
congr b
2,b
4,b
3,b
5 &
congr b
5,b
4,b
3,b
2 &
congr b
3,b
5,b
2,b
4 &
congr b
4,b
2,b
5,b
3 &
congr b
5,b
3,b
4,b
2 ) )
theorem Th90: :: SEMI_AF1:90
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 &
congr b
3,b
6,b
5,b
7 implies
congr b
2,b
6,b
4,b
7 )
theorem Th91: :: SEMI_AF1:91
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 &
congr b
6,b
3,b
4,b
7 implies
congr b
2,b
6,b
7,b
5 )
theorem Th92: :: SEMI_AF1:92
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
congr b
2,b
3,b
3,b
4 &
congr b
5,b
3,b
3,b
6 implies
congr b
2,b
5,b
6,b
4 )
by Th91;
theorem Th93: :: SEMI_AF1:93
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
congr b
2,b
3,b
4,b
5 &
congr b
6,b
3,b
4,b
7 implies b
2,b
6 // b
5,b
7 )
theorem Th94: :: SEMI_AF1:94
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
congr b
2,b
3,b
3,b
4 &
congr b
5,b
3,b
3,b
6 implies b
2,b
5 // b
4,b
6 )
by Th93;
definition
let c
1 be
Semi_Affine_Space;
let c
2, c
3, c
4 be
Element of c
1;
func sum c
2,c
3,c
4 -> Element of a
1 means :
Def5:
:: SEMI_AF1:def 5
congr a
4,a
2,a
3,a
5;
correctness
existence
ex b1 being Element of c1 st congr c4,c2,c3,b1;
uniqueness
for b1, b2 being Element of c1 holds
( congr c4,c2,c3,b1 & congr c4,c2,c3,b2 implies b1 = b2 );
by Th81, Th82;
end;
:: deftheorem Def5 defines sum SEMI_AF1:def 5 :
definition
let c
1 be
Semi_Affine_Space;
let c
2, c
3 be
Element of c
1;
func opposite c
2,c
3 -> Element of a
1 means :
Def6:
:: SEMI_AF1:def 6
sum a
2,a
4,a
3 = a
3;
existence
ex b1 being Element of c1 st sum c2,b1,c3 = c3
uniqueness
for b1, b2 being Element of c1 holds
( sum c2,b1,c3 = c3 & sum c2,b2,c3 = c3 implies b1 = b2 )
end;
:: deftheorem Def6 defines opposite SEMI_AF1:def 6 :
definition
let c
1 be
Semi_Affine_Space;
let c
2, c
3, c
4 be
Element of c
1;
func diff c
2,c
3,c
4 -> Element of a
1 equals :: SEMI_AF1:def 7
sum a
2,
(opposite a3,a4),a
4;
correctness
coherence
sum c2,(opposite c3,c4),c4 is Element of c1;
;
end;
:: deftheorem Def7 defines diff SEMI_AF1:def 7 :
theorem Th95: :: SEMI_AF1:95
canceled;
theorem Th96: :: SEMI_AF1:96
canceled;
theorem Th97: :: SEMI_AF1:97
canceled;
theorem Th98: :: SEMI_AF1:98
canceled;
theorem Th99: :: SEMI_AF1:99
theorem Th100: :: SEMI_AF1:100
theorem Th101: :: SEMI_AF1:101
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
sum (sum b2,b3,b4),b
5,b
4 = sum b
2,
(sum b3,b5,b4),b
4
theorem Th102: :: SEMI_AF1:102
theorem Th103: :: SEMI_AF1:103
theorem Th104: :: SEMI_AF1:104
theorem Th105: :: SEMI_AF1:105
canceled;
theorem Th106: :: SEMI_AF1:106
theorem Th107: :: SEMI_AF1:107
theorem Th108: :: SEMI_AF1:108
theorem Th109: :: SEMI_AF1:109
theorem Th110: :: SEMI_AF1:110
theorem Th111: :: SEMI_AF1:111
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,b
3 // b
4,b
5 implies b
2,b
3 // sum b
2,b
4,b
6,
sum b
3,b
5,b
6 )
theorem Th112: :: SEMI_AF1:112
canceled;
theorem Th113: :: SEMI_AF1:113
theorem Th114: :: SEMI_AF1:114
theorem Th115: :: SEMI_AF1:115
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2,
diff b
3,b
4,b
2,
diff b
5,b
6,b
2 is_collinear iff b
4,b
3 // b
6,b
5 )
definition
let c
1 be
Semi_Affine_Space;
let c
2, c
3, c
4, c
5, c
6 be
Element of c
1;
pred trap c
2,c
3,c
4,c
5,c
6 means :
Def8:
:: SEMI_AF1:def 8
( not a
6,a
2,a
4 is_collinear & a
6,a
2,a
3 is_collinear & a
6,a
4,a
5 is_collinear & a
2,a
4 // a
3,a
5 );
end;
:: deftheorem Def8 defines trap SEMI_AF1:def 8 :
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
trap b
2,b
3,b
4,b
5,b
6 iff ( not b
6,b
2,b
4 is_collinear & b
6,b
2,b
3 is_collinear & b
6,b
4,b
5 is_collinear & b
2,b
4 // b
3,b
5 ) );
:: deftheorem Def9 defines qtrap SEMI_AF1:def 9 :
theorem Th116: :: SEMI_AF1:116
canceled;
theorem Th117: :: SEMI_AF1:117
canceled;
theorem Th118: :: SEMI_AF1:118
theorem Th119: :: SEMI_AF1:119
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
trap b
2,b
3,b
4,b
5,b
6 &
trap b
2,b
3,b
4,b
7,b
6 implies b
5 = b
7 )
theorem Th120: :: SEMI_AF1:120
theorem Th121: :: SEMI_AF1:121
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
trap b
2,b
3,b
4,b
5,b
6 implies
trap b
4,b
5,b
2,b
3,b
6 )
theorem Th122: :: SEMI_AF1:122
theorem Th123: :: SEMI_AF1:123
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( b
2 <> b
3 &
trap b
4,b
3,b
5,b
6,b
2 & b
2,b
3,b
6 is_collinear )
theorem Th124: :: SEMI_AF1:124
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 &
trap b
4,b
3,b
5,b
6,b
2 implies
trap b
3,b
4,b
6,b
5,b
2 )
theorem Th125: :: SEMI_AF1:125
theorem Th126: :: SEMI_AF1:126
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
(
trap b
2,b
3,b
4,b
5,b
6 &
trap b
2,b
3,b
7,b
8,b
6 implies b
4,b
7 // b
5,b
8 )
theorem Th127: :: SEMI_AF1:127
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
(
trap b
2,b
3,b
4,b
5,b
6 &
trap b
2,b
3,b
7,b
8,b
6 & not b
6,b
4,b
7 is_collinear implies
trap b
4,b
5,b
7,b
8,b
6 )
theorem Th128: :: SEMI_AF1:128
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
(
trap b
2,b
3,b
4,b
5,b
6 &
trap b
2,b
3,b
7,b
8,b
6 &
trap b
4,b
5,b
9,b
10,b
6 implies b
7,b
9 // b
8,b
10 )
theorem Th129: :: SEMI_AF1:129
theorem Th130: :: SEMI_AF1:130
theorem Th131: :: SEMI_AF1:131
theorem Th132: :: SEMI_AF1:132
theorem Th133: :: SEMI_AF1:133
for b
1 being
Semi_Affine_Spacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( not b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear &
qtrap b
2,b
3 & ( for b
6 being
Element of b
1 holds
not
trap b
3,b
5,b
4,b
6,b
2 ) )