:: PARSP_1 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) = b3 - b2
Lemma2:
for b1 being Field
for b2, b3, b4, b5 being Element of b1 holds ((b2 - b3) * (b4 - b5)) - ((b3 - b2) * (b5 - b4)) = 0. b1
Lemma3:
for b1 being Field
for b2, b3, b4, b5 being Element of b1 holds (b2 * (b3 - b3)) - ((b4 - b4) * b5) = 0. b1
Lemma4:
for b1 being Field
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> 0. b1 & (b2 * b3) - (b4 * b5) = 0. b1 & (b2 * b6) - (b7 * b5) = 0. b1 implies (b4 * b6) - (b7 * b3) = 0. b1 )
Lemma5:
for b1 being Field
for b2, b3, b4, b5 being Element of b1 holds
( (b2 * b3) - (b4 * b5) = 0. b1 implies (b5 * b4) - (b3 * b2) = 0. b1 )
Lemma6:
for b1 being Field
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> 0. b1 & (b3 * b4) - (b5 * b2) = 0. b1 & (b3 * b6) - (b7 * b2) = 0. b1 implies (b5 * b6) - (b7 * b4) = 0. b1 )
Lemma7:
for b1 being Field
for b2, b3, b4, b5 being Element of b1 holds (b2 * b3) * (b4 * b5) = ((b4 * b2) * b3) * b5
Lemma8:
for b1 being Field
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( (b2 * b3) - (b4 * b5) = 0. b1 implies (((b2 * b6) * b7) * b3) - (((b4 * b6) * b7) * b5) = 0. b1 )
Lemma9:
for b1 being Field
for b2, b3, b4, b5 being Element of b1 holds (b2 - b3) * (b4 - b5) = (b2 * b4) + ((- (b2 * b5)) + (- (b3 * (b4 - b5))))
Lemma10:
for b1 being Field
for b2, b3, b4, b5 being Element of b1 holds (b2 + b3) + (b4 + b5) = (b2 + (b3 + b4)) + b5
Lemma11:
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
Lemma12:
for b1 being non empty add-associative right_zeroed right_complementable LoopStr
for b2, b3 being Element of b1 holds b2 + b3 = - ((- b3) + (- b2))
Lemma13:
for b1 being Field
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( ((b2 - b3) * (b4 - b5)) - ((b2 - b6) * (b4 - b7)) = 0. b1 implies ((b3 - b2) * (b7 - b5)) - ((b3 - b6) * (b7 - b4)) = 0. b1 )
Lemma14:
for b1 being Field
for b2, b3, b4 being Element of b1 holds b2 - ((b2 + b3) + (- b4)) = b4 - b3
Lemma15:
for b1 being Field
for b2, b3, b4, b5, b6, b7 being Element of b1 holds ((b2 - b3) * (b4 - ((b4 + b5) + (- b6)))) - ((b7 - ((b7 + b3) + (- b2))) * (b6 - b5)) = 0. b1
deffunc H1( Field) -> set = [:the carrier of a1,the carrier of a1,the carrier of a1:];
definition
let c
1 be
Field;
func c3add c
1 -> BinOp of
[:the carrier of a1,the carrier of a1,the carrier of a1:] means :
Def1:
:: PARSP_1:def 1
for b
1, b
2 being
Element of
[:the carrier of a1,the carrier of a1,the carrier of a1:] holds a
2 . b
1,b
2 = [((b1 `1 ) + (b2 `1 )),((b1 `2 ) + (b2 `2 )),((b1 `3 ) + (b2 `3 ))];
existence
ex b1 being BinOp of [:the carrier of c1,the carrier of c1,the carrier of c1:] st
for b2, b3 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds b1 . b2,b3 = [((b2 `1 ) + (b3 `1 )),((b2 `2 ) + (b3 `2 )),((b2 `3 ) + (b3 `3 ))]
uniqueness
for b1, b2 being BinOp of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds
( ( for b3, b4 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds b1 . b3,b4 = [((b3 `1 ) + (b4 `1 )),((b3 `2 ) + (b4 `2 )),((b3 `3 ) + (b4 `3 ))] ) & ( for b3, b4 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds b2 . b3,b4 = [((b3 `1 ) + (b4 `1 )),((b3 `2 ) + (b4 `2 )),((b3 `3 ) + (b4 `3 ))] ) implies b1 = b2 )
end;
:: deftheorem Def1 defines c3add PARSP_1:def 1 :
:: deftheorem Def2 defines + PARSP_1:def 2 :
theorem Th1: :: PARSP_1:1
canceled;
theorem Th2: :: PARSP_1:2
canceled;
theorem Th3: :: PARSP_1:3
theorem Th4: :: PARSP_1:4
for b
1 being
Fieldfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
[b2,b3,b4] + [b5,b6,b7] = [(b2 + b5),(b3 + b6),(b4 + b7)]
definition
let c
1 be
Field;
func c3compl c
1 -> UnOp of
[:the carrier of a1,the carrier of a1,the carrier of a1:] means :
Def3:
:: PARSP_1:def 3
for b
1 being
Element of
[:the carrier of a1,the carrier of a1,the carrier of a1:] holds a
2 . b
1 = [(- (b1 `1 )),(- (b1 `2 )),(- (b1 `3 ))];
existence
ex b1 being UnOp of [:the carrier of c1,the carrier of c1,the carrier of c1:] st
for b2 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds b1 . b2 = [(- (b2 `1 )),(- (b2 `2 )),(- (b2 `3 ))]
uniqueness
for b1, b2 being UnOp of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds
( ( for b3 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds b1 . b3 = [(- (b3 `1 )),(- (b3 `2 )),(- (b3 `3 ))] ) & ( for b3 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] holds b2 . b3 = [(- (b3 `1 )),(- (b3 `2 )),(- (b3 `3 ))] ) implies b1 = b2 )
end;
:: deftheorem Def3 defines c3compl PARSP_1:def 3 :
:: deftheorem Def4 defines - PARSP_1:def 4 :
theorem Th5: :: PARSP_1:5
canceled;
theorem Th6: :: PARSP_1:6
canceled;
theorem Th7: :: PARSP_1:7
:: deftheorem Def5 defines Relation4 PARSP_1:def 5 :
:: deftheorem Def6 defines '||' PARSP_1:def 6 :
:: deftheorem Def7 defines C3 PARSP_1:def 7 :
:: deftheorem Def8 defines 4C3 PARSP_1:def 8 :
definition
let c
1 be
Field;
func PRs c
1 -> set means :
Def9:
:: PARSP_1:def 9
for b
1 being
set holds
( b
1 in a
2 iff ( b
1 in 4C3 a
1 & ex b
2, b
3, b
4, b
5 being
Element of
[:the carrier of a1,the carrier of a1,the carrier of a1:] st
( b
1 = [b2,b3,b4,b5] &
(((b2 `1 ) - (b3 `1 )) * ((b4 `2 ) - (b5 `2 ))) - (((b4 `1 ) - (b5 `1 )) * ((b2 `2 ) - (b3 `2 ))) = 0. a
1 &
(((b2 `1 ) - (b3 `1 )) * ((b4 `3 ) - (b5 `3 ))) - (((b4 `1 ) - (b5 `1 )) * ((b2 `3 ) - (b3 `3 ))) = 0. a
1 &
(((b2 `2 ) - (b3 `2 )) * ((b4 `3 ) - (b5 `3 ))) - (((b4 `2 ) - (b5 `2 )) * ((b2 `3 ) - (b3 `3 ))) = 0. a
1 ) ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 in 4C3 c1 & ex b3, b4, b5, b6 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] st
( b2 = [b3,b4,b5,b6] & (((b3 `1 ) - (b4 `1 )) * ((b5 `2 ) - (b6 `2 ))) - (((b5 `1 ) - (b6 `1 )) * ((b3 `2 ) - (b4 `2 ))) = 0. c1 & (((b3 `1 ) - (b4 `1 )) * ((b5 `3 ) - (b6 `3 ))) - (((b5 `1 ) - (b6 `1 )) * ((b3 `3 ) - (b4 `3 ))) = 0. c1 & (((b3 `2 ) - (b4 `2 )) * ((b5 `3 ) - (b6 `3 ))) - (((b5 `2 ) - (b6 `2 )) * ((b3 `3 ) - (b4 `3 ))) = 0. c1 ) ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 in 4C3 c1 & ex b4, b5, b6, b7 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] st
( b3 = [b4,b5,b6,b7] & (((b4 `1 ) - (b5 `1 )) * ((b6 `2 ) - (b7 `2 ))) - (((b6 `1 ) - (b7 `1 )) * ((b4 `2 ) - (b5 `2 ))) = 0. c1 & (((b4 `1 ) - (b5 `1 )) * ((b6 `3 ) - (b7 `3 ))) - (((b6 `1 ) - (b7 `1 )) * ((b4 `3 ) - (b5 `3 ))) = 0. c1 & (((b4 `2 ) - (b5 `2 )) * ((b6 `3 ) - (b7 `3 ))) - (((b6 `2 ) - (b7 `2 )) * ((b4 `3 ) - (b5 `3 ))) = 0. c1 ) ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 in 4C3 c1 & ex b4, b5, b6, b7 being Element of [:the carrier of c1,the carrier of c1,the carrier of c1:] st
( b3 = [b4,b5,b6,b7] & (((b4 `1 ) - (b5 `1 )) * ((b6 `2 ) - (b7 `2 ))) - (((b6 `1 ) - (b7 `1 )) * ((b4 `2 ) - (b5 `2 ))) = 0. c1 & (((b4 `1 ) - (b5 `1 )) * ((b6 `3 ) - (b7 `3 ))) - (((b6 `1 ) - (b7 `1 )) * ((b4 `3 ) - (b5 `3 ))) = 0. c1 & (((b4 `2 ) - (b5 `2 )) * ((b6 `3 ) - (b7 `3 ))) - (((b6 `2 ) - (b7 `2 )) * ((b4 `3 ) - (b5 `3 ))) = 0. c1 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def9 defines PRs PARSP_1:def 9 :
theorem Th8: :: PARSP_1:8
canceled;
theorem Th9: :: PARSP_1:9
canceled;
theorem Th10: :: PARSP_1:10
canceled;
theorem Th11: :: PARSP_1:11
canceled;
theorem Th12: :: PARSP_1:12
canceled;
theorem Th13: :: PARSP_1:13
:: deftheorem Def10 defines PR PARSP_1:def 10 :
:: deftheorem Def11 defines MPS PARSP_1:def 11 :
theorem Th14: :: PARSP_1:14
canceled;
theorem Th15: :: PARSP_1:15
canceled;
theorem Th16: :: PARSP_1:16
theorem Th17: :: PARSP_1:17
theorem Th18: :: PARSP_1:18
theorem Th19: :: PARSP_1:19
for b
1 being
Fieldfor b
2, b
3, b
4, b
5 being
Element of
(MPS b1) holds
(
[b2,b3,b4,b5] in PR b
1 iff (
[b2,b3,b4,b5] in 4C3 b
1 & 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] &
(((b6 `1 ) - (b7 `1 )) * ((b8 `2 ) - (b9 `2 ))) - (((b8 `1 ) - (b9 `1 )) * ((b6 `2 ) - (b7 `2 ))) = 0. b
1 &
(((b6 `1 ) - (b7 `1 )) * ((b8 `3 ) - (b9 `3 ))) - (((b8 `1 ) - (b9 `1 )) * ((b6 `3 ) - (b7 `3 ))) = 0. b
1 &
(((b6 `2 ) - (b7 `2 )) * ((b8 `3 ) - (b9 `3 ))) - (((b8 `2 ) - (b9 `2 )) * ((b6 `3 ) - (b7 `3 ))) = 0. b
1 ) ) )
by Def9;
theorem Th20: :: PARSP_1:20
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 (
[b2,b3,b4,b5] in 4C3 b
1 & 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] &
(((b6 `1 ) - (b7 `1 )) * ((b8 `2 ) - (b9 `2 ))) - (((b8 `1 ) - (b9 `1 )) * ((b6 `2 ) - (b7 `2 ))) = 0. b
1 &
(((b6 `1 ) - (b7 `1 )) * ((b8 `3 ) - (b9 `3 ))) - (((b8 `1 ) - (b9 `1 )) * ((b6 `3 ) - (b7 `3 ))) = 0. b
1 &
(((b6 `2 ) - (b7 `2 )) * ((b8 `3 ) - (b9 `3 ))) - (((b8 `2 ) - (b9 `2 )) * ((b6 `3 ) - (b7 `3 ))) = 0. b
1 ) ) )
theorem Th21: :: PARSP_1:21
theorem Th22: :: PARSP_1:22
theorem Th23: :: PARSP_1:23
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] &
(((b6 `1 ) - (b7 `1 )) * ((b8 `2 ) - (b9 `2 ))) - (((b8 `1 ) - (b9 `1 )) * ((b6 `2 ) - (b7 `2 ))) = 0. b
1 &
(((b6 `1 ) - (b7 `1 )) * ((b8 `3 ) - (b9 `3 ))) - (((b8 `1 ) - (b9 `1 )) * ((b6 `3 ) - (b7 `3 ))) = 0. b
1 &
(((b6 `2 ) - (b7 `2 )) * ((b8 `3 ) - (b9 `3 ))) - (((b8 `2 ) - (b9 `2 )) * ((b6 `3 ) - (b7 `3 ))) = 0. b
1 ) )
theorem Th24: :: PARSP_1:24
theorem Th25: :: PARSP_1:25
theorem Th26: :: PARSP_1:26
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
4,b
5 & b
2,b
3 '||' b
6,b
7 & not b
4,b
5 '||' b
6,b
7 & not b
2 = b
3 )
theorem Th27: :: PARSP_1:27
theorem Th28: :: PARSP_1:28
definition
let c
1 be non
empty ParStr ;
attr a
1 is
ParSp-like means :
Def12:
:: PARSP_1:def 12
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of a
1 holds
( b
1,b
2 '||' b
2,b
1 & b
1,b
2 '||' b
3,b
3 & not ( b
1,b
2 '||' b
5,b
6 & b
1,b
2 '||' b
7,b
8 & not b
5,b
6 '||' b
7,b
8 & not b
1 = b
2 ) & ( b
1,b
2 '||' b
1,b
3 implies b
2,b
1 '||' b
2,b
3 ) & ex b
9 being
Element of a
1 st
( b
1,b
2 '||' b
3,b
9 & b
1,b
3 '||' b
2,b
9 ) );
end;
:: deftheorem Def12 defines ParSp-like PARSP_1:def 12 :
for b
1 being non
empty ParStr holds
( b
1 is
ParSp-like iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
2,b
3 '||' b
3,b
2 & b
2,b
3 '||' b
4,b
4 & not ( b
2,b
3 '||' b
6,b
7 & b
2,b
3 '||' b
8,b
9 & not b
6,b
7 '||' b
8,b
9 & not b
2 = b
3 ) & ( b
2,b
3 '||' b
2,b
4 implies b
3,b
2 '||' b
3,b
4 ) & ex b
10 being
Element of b
1 st
( b
2,b
3 '||' b
4,b
10 & b
2,b
4 '||' b
3,b
10 ) ) );
theorem Th29: :: PARSP_1:29
canceled;
theorem Th30: :: PARSP_1:30
canceled;
theorem Th31: :: PARSP_1:31
canceled;
theorem Th32: :: PARSP_1:32
canceled;
theorem Th33: :: PARSP_1:33
canceled;
theorem Th34: :: PARSP_1:34
canceled;
theorem Th35: :: PARSP_1:35
theorem Th36: :: PARSP_1:36
for b
1 being
ParSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 '||' b
4,b
5 implies b
4,b
5 '||' b
2,b
3 )
theorem Th37: :: PARSP_1:37
theorem Th38: :: PARSP_1:38
for b
1 being
ParSpfor 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 )
theorem Th39: :: PARSP_1:39
for b
1 being
ParSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 '||' b
4,b
5 implies b
2,b
3 '||' b
5,b
4 )
theorem Th40: :: PARSP_1:40
for b
1 being
ParSpfor 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 Th41: :: PARSP_1:41
for b
1 being
ParSpfor 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 Th42: :: PARSP_1:42
for b
1 being
ParSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( not ( not b
2 = b
3 & not b
4 = b
5 & not ( b
2 = b
4 & b
3 = b
5 ) & not ( b
2 = b
5 & b
3 = b
4 ) ) implies b
2,b
3 '||' b
4,b
5 )
by Def12, Th35, Th37;
theorem Th43: :: PARSP_1:43
for b
1 being
ParSpfor 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 Th44: :: PARSP_1:44
theorem Th45: :: PARSP_1:45
theorem Th46: :: PARSP_1:46
canceled;
theorem Th47: :: PARSP_1:47
for b
1 being
ParSpfor 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
4 '||' b
2,b
3 & not b
3,b
2 '||' b
2,b
4 & not b
2,b
3 '||' b
4,b
2 & not b
2,b
4 '||' b
3,b
2 & not b
3,b
2 '||' b
4,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
2,b
3 '||' b
3,b
4 & not b
3,b
2 '||' b
4,b
3 & not b
3,b
4 '||' b
3,b
2 & not b
3,b
2 '||' b
4,b
3 & not b
4,b
3 '||' 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
2 '||' b
4,b
3 & not b
2,b
4 '||' b
4,b
3 & not b
4,b
2 '||' b
3,b
4 & not b
2,b
4 '||' b
3,b
4 & not b
4,b
3 '||' b
4,b
2 & not b
3,b
4 '||' b
4,b
2 & not b
4,b
3 '||' b
2,b
4 & not b
3,b
4 '||' b
2,b
4 ) )
theorem Th48: :: PARSP_1:48
for b
1 being
ParSpfor 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 Th49: :: PARSP_1:49
for b
1 being
ParSpfor 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 Th50: :: PARSP_1:50
for b
1 being
ParSpfor 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 Th51: :: PARSP_1:51
for b
1 being
ParSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( not b
2,b
3 '||' b
2,b
4 & b
2,b
4 '||' b
2,b
5 & b
3,b
4 '||' b
3,b
5 implies b
4 = b
5 )
theorem Th52: :: PARSP_1:52
for b
1 being
ParSpfor 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 Th53: :: PARSP_1:53
for b
1 being
ParSpfor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3 '||' b
2,b
4 & b
2,b
3 '||' b
2,b
5 implies b
2,b
3 '||' b
4,b
5 )
theorem Th54: :: PARSP_1:54
for b
1 being
ParSp holds
( ( for b
2, b
3 being
Element of b
1 holds b
2 = b
3 ) implies for b
2, b
3, b
4, b
5 being
Element of b
1 holds b
2,b
3 '||' b
4,b
5 )
theorem Th55: :: PARSP_1:55
for b
1 being
ParSp holds
( ex b
2, b
3 being
Element of b
1 st
( b
2 <> b
3 & ( for b
4 being
Element of b
1 holds b
2,b
3 '||' b
2,b
4 ) ) implies for b
2, b
3, b
4, b
5 being
Element of b
1 holds b
2,b
3 '||' b
4,b
5 )
theorem Th56: :: PARSP_1:56
for b
1 being
ParSpfor 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 )