:: PARSP_2 semantic presentation
Lemma1:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Element of b1 holds
( b2 - b3 = 0. b1 implies b2 = b3 )
Lemma2:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Element of b1 holds
( ( b2 + (- b3) = 0. b1 implies b2 = b3 ) & ( b2 = b3 implies b2 + (- b3) = 0. b1 ) & ( b2 - b3 = 0. b1 implies b2 = b3 ) & ( b2 = b3 implies b2 - b3 = 0. b1 ) )
theorem Th1: :: PARSP_2:1
Lemma4:
for b1 being Field
for b2, b3, b4, b5 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] holds
( ( (((b2 `1 ) - (b3 `1 )) * ((b4 `2 ) - (b5 `2 ))) - (((b4 `1 ) - (b5 `1 )) * ((b2 `2 ) - (b3 `2 ))) = 0. b1 & (((b2 `1 ) - (b3 `1 )) * ((b4 `3 ) - (b5 `3 ))) - (((b4 `1 ) - (b5 `1 )) * ((b2 `3 ) - (b3 `3 ))) = 0. b1 & (((b2 `2 ) - (b3 `2 )) * ((b4 `3 ) - (b5 `3 ))) - (((b4 `2 ) - (b5 `2 )) * ((b2 `3 ) - (b3 `3 ))) = 0. b1 ) iff not ( ( for b6 being Element of b1 holds
not ( b6 * ((b2 `1 ) - (b3 `1 )) = (b4 `1 ) - (b5 `1 ) & b6 * ((b2 `2 ) - (b3 `2 )) = (b4 `2 ) - (b5 `2 ) & b6 * ((b2 `3 ) - (b3 `3 )) = (b4 `3 ) - (b5 `3 ) ) ) & not ( (b2 `1 ) - (b3 `1 ) = 0. b1 & (b2 `2 ) - (b3 `2 ) = 0. b1 & (b2 `3 ) - (b3 `3 ) = 0. b1 ) ) )
theorem Th2: :: PARSP_2:2
for b
1 being
Fieldfor b
2, b
3, b
4, b
5 being
Element of
(MPS b1) holds
( b
2,b
3 '||' b
4,b
5 iff ex b
6, b
7, b
8, b
9 being
Element of
[:the carrier of b1,the carrier of b1,the carrier of b1:] st
(
[b2,b3,b4,b5] = [b6,b7,b8,b9] & not ( ( for b
10 being
Element of b
1 holds
not ( b
10 * ((b6 `1 ) - (b7 `1 )) = (b8 `1 ) - (b9 `1 ) & b
10 * ((b6 `2 ) - (b7 `2 )) = (b8 `2 ) - (b9 `2 ) & b
10 * ((b6 `3 ) - (b7 `3 )) = (b8 `3 ) - (b9 `3 ) ) ) & not (
(b6 `1 ) - (b7 `1 ) = 0. b
1 &
(b6 `2 ) - (b7 `2 ) = 0. b
1 &
(b6 `3 ) - (b7 `3 ) = 0. b
1 ) ) ) )
theorem Th3: :: PARSP_2:3
for b
1 being
Fieldfor b
2, b
3, b
4 being
Element of
(MPS b1)for b
5, b
6, b
7 being
Element of
[:the carrier of b1,the carrier of b1,the carrier of b1:] holds
( not b
2,b
3 '||' b
2,b
4 &
[b2,b3,b2,b4] = [b5,b6,b5,b7] implies ( b
5 <> b
6 & b
5 <> b
7 & b
6 <> b
7 ) )
theorem Th4: :: PARSP_2:4
for b
1 being
Fieldfor b
2, b
3, b
4 being
Element of
(MPS b1)for b
5, b
6, b
7 being
Element of
[:the carrier of b1,the carrier of b1,the carrier of b1:]for b
8, b
9 being
Element of b
1 holds
( not b
2,b
3 '||' b
2,b
4 &
[b2,b3,b2,b4] = [b5,b6,b5,b7] & b
8 * ((b5 `1 ) - (b6 `1 )) = b
9 * ((b5 `1 ) - (b7 `1 )) & b
8 * ((b5 `2 ) - (b6 `2 )) = b
9 * ((b5 `2 ) - (b7 `2 )) & b
8 * ((b5 `3 ) - (b6 `3 )) = b
9 * ((b5 `3 ) - (b7 `3 )) implies ( b
8 = 0. b
1 & b
9 = 0. b
1 ) )
Lemma8:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3, b4 being Element of b1 holds (b3 + b2) - (b4 + b2) = b3 - b4
Lemma9:
for b1 being Field
for b2, b3, b4 being Element of b1 holds (b2 - b3) - (b4 - b3) = b2 - b4
Lemma10:
for b1 being Field
for b2, b3, b4, b5, b6 being Element of b1 holds
( (b2 * (b3 - b4)) - (b5 * (b3 - b6)) = b6 - b4 implies (b2 + (- (1. b1))) * (b3 - b4) = (b5 + (- (1. b1))) * (b3 - b6) )
Lemma11:
for b1 being Field
for b2, b3, b4, b5 being Element of b1 holds
( b2 - b3 = b4 - b5 implies b5 = (b3 + b4) - b2 )
theorem Th5: :: PARSP_2:5
for b
1 being
Fieldfor b
2, b
3, b
4, b
5 being
Element of
(MPS b1)for b
6, b
7, b
8, b
9 being
Element of
[:the carrier of b1,the carrier of b1,the carrier of b1:] holds
( 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 &
[b2,b3,b4,b5] = [b6,b7,b8,b9] implies ( b
9 `1 = ((b7 `1 ) + (b8 `1 )) - (b6 `1 ) & b
9 `2 = ((b7 `2 ) + (b8 `2 )) - (b6 `2 ) & b
9 `3 = ((b7 `3 ) + (b8 `3 )) - (b6 `3 ) ) )
Lemma13:
for b1 being Field
for b2, b3, b4 being Element of b1 holds
( (b2 * b3) - (b2 * b4) = b4 + b3 implies (b2 - (1. b1)) * b3 = (b2 + (1. b1)) * b4 )
Lemma14:
for b1 being Field
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 * (b3 - b4) = b5 - b6 & b6 = (b3 + b4) - b5 implies (b2 - (1. b1)) * (b5 - b4) = (b2 + (1. b1)) * (b5 - b3) )
Lemma15:
for b1 being Field
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 = (b3 + b4) - b5 & b6 = (b3 + b7) - b5 implies (1. b1) * (b4 - b7) = b2 - b6 )
theorem Th6: :: PARSP_2:6
theorem Th7: :: PARSP_2:7
for b
1 being
Fieldfor b
2, b
3, b
4, b
5 being
Element of
(MPS b1) holds
(
(1. b1) + (1. b1) <> 0. b
1 & b
2,b
3 '||' b
4,b
5 & b
4,b
2 '||' b
3,b
5 & b
4,b
3 '||' b
2,b
5 implies b
4,b
2 '||' b
4,b
3 )
theorem Th8: :: PARSP_2:8
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
(MPS b1) holds
( not b
2,b
3 '||' b
2,b
4 & not b
2,b
3 '||' b
2,b
5 & b
2,b
3 '||' b
4,b
6 & b
2,b
3 '||' b
5,b
7 & b
2,b
4 '||' b
3,b
6 & b
2,b
5 '||' b
3,b
7 implies b
4,b
5 '||' b
6,b
7 )
definition
let c
1 be
ParSp;
attr a
1 is
FanodesSp-like means :
Def1:
:: PARSP_2:def 1
( 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, b
4 being
Element of a
1 holds
( b
2,b
3 '||' b
1,b
4 & b
1,b
2 '||' b
3,b
4 & b
1,b
3 '||' b
2,b
4 implies b
1,b
2 '||' b
1,b
3 ) ) & ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( not b
1,b
4 '||' b
1,b
2 & not b
1,b
4 '||' b
1,b
3 & b
1,b
4 '||' b
2,b
5 & b
1,b
4 '||' b
3,b
6 & b
1,b
2 '||' b
4,b
5 & b
1,b
3 '||' b
4,b
6 implies b
2,b
3 '||' b
5,b
6 ) ) );
end;
:: deftheorem Def1 defines FanodesSp-like PARSP_2:def 1 :
for b
1 being
ParSp holds
( b
1 is
FanodesSp-like iff ( 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, b
5 being
Element of b
1 holds
( b
3,b
4 '||' b
2,b
5 & b
2,b
3 '||' b
4,b
5 & b
2,b
4 '||' b
3,b
5 implies b
2,b
3 '||' b
2,b
4 ) ) & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
5 '||' b
2,b
3 & not b
2,b
5 '||' b
2,b
4 & b
2,b
5 '||' b
3,b
6 & b
2,b
5 '||' b
4,b
7 & b
2,b
3 '||' b
5,b
6 & b
2,b
4 '||' b
5,b
7 implies b
3,b
4 '||' b
6,b
7 ) ) ) );
theorem Th9: :: PARSP_2:9
canceled;
theorem Th10: :: PARSP_2:10
canceled;
theorem Th11: :: PARSP_2:11
canceled;
theorem Th12: :: PARSP_2:12
canceled;
theorem Th13: :: PARSP_2:13
:: deftheorem Def2 defines is_collinear PARSP_2:def 2 :
theorem Th14: :: PARSP_2:14
canceled;
theorem Th15: :: PARSP_2:15
for b
1 being
FanodesSpfor 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
4,b
3,b
2 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 ) )
theorem Th16: :: PARSP_2:16
canceled;
theorem Th17: :: PARSP_2:17
for b
1 being
FanodesSpfor 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 Th18: :: PARSP_2:18
theorem Th19: :: PARSP_2:19
for b
1 being
FanodesSpfor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 & b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
3,b
6 is_collinear implies b
4,b
5,b
6 is_collinear )
theorem Th20: :: PARSP_2:20
theorem Th21: :: PARSP_2:21
theorem Th22: :: PARSP_2:22
theorem Th23: :: PARSP_2:23
for b
1 being
FanodesSpfor 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
2,b
3,b
6 is_collinear & b
4,b
5,b
6 is_collinear )
theorem Th24: :: PARSP_2:24
theorem Th25: :: PARSP_2:25
for b
1 being
FanodesSpfor 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 Th26: :: PARSP_2:26
for b
1 being
FanodesSpfor 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 Th27: :: PARSP_2:27
theorem Th28: :: PARSP_2:28
theorem Th29: :: PARSP_2:29
for b
1 being
FanodesSpfor 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
7 implies b
6 = b
7 )
theorem Th30: :: PARSP_2:30
theorem Th31: :: PARSP_2:31
definition
let c
1 be
FanodesSp;
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:
:: PARSP_2: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 PARSP_2:def 3 :
for b
1 being
FanodesSpfor 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 Th32: :: PARSP_2:32
canceled;
theorem Th33: :: PARSP_2:33
canceled;
theorem Th34: :: PARSP_2:34
theorem Th35: :: PARSP_2:35
for b
1 being
FanodesSpfor 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 Th36: :: PARSP_2:36
for b
1 being
FanodesSpfor 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 & not b
2,b
4,b
3 is_collinear & not b
3,b
2,b
4 is_collinear & not b
3,b
4,b
2 is_collinear & not b
4,b
2,b
3 is_collinear & not b
4,b
3,b
2 is_collinear & not b
3,b
5,b
2 is_collinear & not b
2,b
3,b
5 is_collinear & not b
2,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
4,b
2,b
5 is_collinear & not b
2,b
4,b
5 is_collinear & not b
2,b
5,b
4 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
3,b
4,b
5 is_collinear & not b
3,b
5,b
4 is_collinear & not b
4,b
3,b
5 is_collinear & not b
4,b
5,b
3 is_collinear ) )
theorem Th37: :: PARSP_2:37
for b
1 being
FanodesSpfor 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 Th38: :: PARSP_2:38
theorem Th39: :: PARSP_2:39
theorem Th40: :: PARSP_2:40
theorem Th41: :: PARSP_2:41
for b
1 being
FanodesSpfor 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 &
parallelogram b
5,b
4,b
3,b
2 ) )
theorem Th42: :: PARSP_2:42
theorem Th43: :: PARSP_2:43
for b
1 being
FanodesSpfor 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 Th44: :: PARSP_2:44
theorem Th45: :: PARSP_2:45
theorem Th46: :: PARSP_2:46
theorem Th47: :: PARSP_2:47
for b
1 being
FanodesSpfor 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 Th48: :: PARSP_2:48
for b
1 being
FanodesSpfor 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 Th49: :: PARSP_2:49
for b
1 being
FanodesSpfor 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 Th50: :: PARSP_2:50
for b
1 being
FanodesSpfor 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 )
theorem Th51: :: PARSP_2:51
theorem Th52: :: PARSP_2:52
definition
let c
1 be
FanodesSp;
let c
2, c
3, c
4, c
5 be
Element of c
1;
pred c
2,c
3 congr c
4,c
5 means :
Def4:
:: PARSP_2: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 PARSP_2:def 4 :
for b
1 being
FanodesSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 congr 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 Th53: :: PARSP_2:53
canceled;
theorem Th54: :: PARSP_2:54
canceled;
theorem Th55: :: PARSP_2:55
theorem Th56: :: PARSP_2:56
theorem Th57: :: PARSP_2:57
theorem Th58: :: PARSP_2:58
theorem Th59: :: PARSP_2:59
theorem Th60: :: PARSP_2:60
for b
1 being
FanodesSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 congr b
4,b
5 & not b
2,b
3,b
4 is_collinear implies
parallelogram b
2,b
3,b
4,b
5 )
theorem Th61: :: PARSP_2:61
theorem Th62: :: PARSP_2:62
for b
1 being
FanodesSpfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 congr 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 Th63: :: PARSP_2:63
theorem Th64: :: PARSP_2:64
theorem Th65: :: PARSP_2:65
canceled;
theorem Th66: :: PARSP_2:66
theorem Th67: :: PARSP_2:67
for b
1 being
FanodesSpfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 congr b
4,b
5 & b
2,b
3 congr b
6,b
7 implies b
4,b
5 congr b
6,b
7 )
theorem Th68: :: PARSP_2:68
theorem Th69: :: PARSP_2:69