:: DIRAF semantic presentation
definition
let c
1 be non
empty set ;
let c
2 be
Relation of
[:c1,c1:];
func lambda c
2 -> Relation of
[:a1,a1:] means :
Def1:
:: DIRAF:def 1
for b
1, b
2, b
3, b
4 being
Element of a
1 holds
(
[[b1,b2],[b3,b4]] in a
3 iff (
[[b1,b2],[b3,b4]] in a
2 or
[[b1,b2],[b4,b3]] in a
2 ) );
existence
ex b1 being Relation of [:c1,c1:] st
for b2, b3, b4, b5 being Element of c1 holds
( [[b2,b3],[b4,b5]] in b1 iff ( [[b2,b3],[b4,b5]] in c2 or [[b2,b3],[b5,b4]] in c2 ) )
uniqueness
for b1, b2 being Relation of [:c1,c1:] holds
( ( for b3, b4, b5, b6 being Element of c1 holds
( [[b3,b4],[b5,b6]] in b1 iff ( [[b3,b4],[b5,b6]] in c2 or [[b3,b4],[b6,b5]] in c2 ) ) ) & ( for b3, b4, b5, b6 being Element of c1 holds
( [[b3,b4],[b5,b6]] in b2 iff ( [[b3,b4],[b5,b6]] in c2 or [[b3,b4],[b6,b5]] in c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines lambda DIRAF:def 1 :
for b
1 being non
empty set for b
2, b
3 being
Relation of
[:b1,b1:] holds
( b
3 = lambda b
2 iff for b
4, b
5, b
6, b
7 being
Element of b
1 holds
(
[[b4,b5],[b6,b7]] in b
3 iff (
[[b4,b5],[b6,b7]] in b
2 or
[[b4,b5],[b7,b6]] in b
2 ) ) );
:: deftheorem Def2 defines Lambda DIRAF:def 2 :
theorem Th1: :: DIRAF:1
canceled;
theorem Th2: :: DIRAF:2
canceled;
theorem Th3: :: DIRAF:3
canceled;
theorem Th4: :: DIRAF:4
Lemma3:
for b1 being OAffinSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3 // b4,b5 implies b4,b5 // b2,b3 )
theorem Th5: :: DIRAF:5
for b
1 being
OAffinSpacefor 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
5,b
4 & b
4,b
5 // b
2,b
3 & b
5,b
4 // b
3,b
2 ) )
theorem Th6: :: DIRAF:6
for b
1 being
OAffinSpacefor 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 Th7: :: DIRAF:7
theorem Th8: :: DIRAF:8
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2,b
3 // b
4,b
5 & b
2,b
3 // b
5,b
4 & not b
2 = b
3 & not b
4 = b
5 )
theorem Th9: :: DIRAF:9
for b
1 being
OAffinSpacefor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3 // b
2,b
4 iff ( b
2,b
3 // b
3,b
4 or b
2,b
4 // b
4,b
3 ) )
:: deftheorem Def3 defines Mid DIRAF:def 3 :
theorem Th10: :: DIRAF:10
canceled;
theorem Th11: :: DIRAF:11
theorem Th12: :: DIRAF:12
theorem Th13: :: DIRAF:13
theorem Th14: :: DIRAF:14
theorem Th15: :: DIRAF:15
theorem Th16: :: DIRAF:16
theorem Th17: :: DIRAF:17
theorem Th18: :: DIRAF:18
theorem Th19: :: DIRAF:19
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2 <> b
3 &
Mid b
2,b
3,b
4 &
Mid b
2,b
3,b
5 & not
Mid b
3,b
4,b
5 & not
Mid b
3,b
5,b
4 )
theorem Th20: :: DIRAF:20
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2 <> b
3 &
Mid b
2,b
3,b
4 &
Mid b
2,b
3,b
5 & not
Mid b
2,b
4,b
5 & not
Mid b
2,b
5,b
4 )
theorem Th21: :: DIRAF:21
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
not (
Mid b
2,b
3,b
4 &
Mid b
2,b
5,b
4 & not
Mid b
2,b
3,b
5 & not
Mid b
2,b
5,b
3 )
:: deftheorem Def4 defines '||' DIRAF:def 4 :
theorem Th22: :: DIRAF:22
canceled;
theorem Th23: :: DIRAF:23
theorem Th24: :: DIRAF:24
theorem Th25: :: DIRAF:25
Lemma16:
for b1 being OAffinSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> b3 & b2,b3 '||' b4,b5 & b2,b3 '||' b6,b7 implies b4,b5 '||' b6,b7 )
theorem Th26: :: DIRAF:26
theorem Th27: :: DIRAF:27
for b
1 being
OAffinSpacefor 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 & b
3,b
2 '||' b
4,b
5 & b
3,b
2 '||' b
5,b
4 & b
4,b
5 '||' b
2,b
3 & b
4,b
5 '||' b
3,b
2 & b
5,b
4 '||' b
2,b
3 & b
5,b
4 '||' b
3,b
2 ) )
theorem Th28: :: DIRAF:28
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & not ( not ( b
2,b
3 '||' b
4,b
5 & b
2,b
3 '||' b
6,b
7 ) & not ( b
2,b
3 '||' b
4,b
5 & b
6,b
7 '||' b
2,b
3 ) & not ( b
4,b
5 '||' b
2,b
3 & b
6,b
7 '||' b
2,b
3 ) & not ( 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 Th29: :: DIRAF:29
theorem Th30: :: DIRAF:30
theorem Th31: :: DIRAF:31
theorem Th32: :: DIRAF:32
:: deftheorem Def5 defines LIN DIRAF:def 5 :
theorem Th33: :: DIRAF:33
canceled;
theorem Th34: :: DIRAF:34
theorem Th35: :: DIRAF:35
for b
1 being
OAffinSpacefor b
2, b
3, b
4 being
Element of b
1 holds
not ( b
2,b
3,b
4 is_collinear & not
Mid b
2,b
3,b
4 & not
Mid b
3,b
2,b
4 & not
Mid b
2,b
4,b
3 )
Lemma25:
for b1 being OAffinSpace
for b2, b3, b4 being Element of b1 holds
( b2,b3,b4 is_collinear implies ( b2,b4,b3 is_collinear & b3,b2,b4 is_collinear ) )
theorem Th36: :: DIRAF:36
for b
1 being
OAffinSpacefor 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 Th37: :: DIRAF:37
theorem Th38: :: DIRAF:38
for b
1 being
OAffinSpacefor 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 Th39: :: DIRAF:39
theorem Th40: :: DIRAF:40
theorem Th41: :: DIRAF:41
for b
1 being
OAffinSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 & b
4,b
5,b
2 is_collinear & b
4,b
5,b
3 is_collinear & b
2,b
3,b
6 is_collinear implies b
4,b
5,b
6 is_collinear )
theorem Th42: :: DIRAF:42
theorem Th43: :: DIRAF:43
theorem Th44: :: DIRAF:44
canceled;
theorem Th45: :: DIRAF:45
for b
1 being
OAffinSpacefor b
2 being non
empty AffinStruct holds
( b
2 = Lambda b
1 implies for b
3, b
4, b
5, b
6 being
Element of b
1for b
7, b
8, b
9, b
10 being
Element of b
2 holds
( b
3 = b
7 & b
4 = b
8 & b
5 = b
9 & b
6 = b
10 implies ( b
7,b
8 // b
9,b
10 iff b
3,b
4 '||' b
5,b
6 ) ) )
theorem Th46: :: DIRAF:46
for b
1 being
OAffinSpacefor b
2 being non
empty AffinStruct holds
( b
2 = Lambda b
1 implies ( not for b
3, b
4 being
Element of b
2 holds not b
3 <> b
4 & ( for b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
2 holds
( b
3,b
4 // b
4,b
3 & b
3,b
4 // b
5,b
5 & ( b
3 <> b
4 & b
3,b
4 // b
5,b
6 & b
3,b
4 // b
7,b
8 implies b
5,b
6 // b
7,b
8 ) & ( b
3,b
4 // b
3,b
5 implies b
4,b
3 // b
4,b
5 ) ) ) & not for b
3, b
4, b
5 being
Element of b
2 holds b
3,b
4 // b
3,b
5 & ( for b
3, b
4, b
5 being
Element of b
2 holds
ex b
6 being
Element of b
2 st
( b
3,b
5 // b
4,b
6 & b
4 <> b
6 ) ) & ( for b
3, b
4, b
5 being
Element of b
2 holds
ex b
6 being
Element of b
2 st
( b
3,b
4 // b
5,b
6 & b
3,b
5 // b
4,b
6 ) ) & ( for b
3, b
4, b
5, b
6 being
Element of b
2 holds
not ( b
5,b
3 // b
3,b
6 & b
3 <> b
5 & ( for b
7 being
Element of b
2 holds
not ( b
4,b
3 // b
3,b
7 & b
4,b
5 // b
6,b
7 ) ) ) ) ) )
definition
let c
1 be non
empty AffinStruct ;
canceled;attr a
1 is
AffinSpace-like means :
Def7:
:: DIRAF:def 7
( ( for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
2,b
1 & b
1,b
2 // b
3,b
3 & ( 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 ) & ( 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
3 // b
2,b
4 & b
2 <> b
4 ) ) & ( 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, b
3, b
4 being
Element of a
1 holds
not ( b
3,b
1 // b
1,b
4 & b
1 <> b
3 & ( for b
5 being
Element of a
1 holds
not ( b
2,b
1 // b
1,b
5 & b
2,b
3 // b
4,b
5 ) ) ) ) );
end;
:: deftheorem Def6 DIRAF:def 6 :
canceled;
:: deftheorem Def7 defines AffinSpace-like DIRAF:def 7 :
for b
1 being non
empty AffinStruct holds
( b
1 is
AffinSpace-like iff ( ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
3,b
2 & b
2,b
3 // b
4,b
4 & ( 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 ) & ( 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
4 // b
3,b
5 & b
3 <> b
5 ) ) & ( 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, b
4, b
5 being
Element of b
1 holds
not ( b
4,b
2 // b
2,b
5 & b
2 <> b
4 & ( for b
6 being
Element of b
1 holds
not ( b
3,b
2 // b
2,b
6 & b
3,b
4 // b
5,b
6 ) ) ) ) ) );
theorem Th47: :: DIRAF:47
for b
1 being
AffinSpace holds
( not for b
2, b
3 being
Element of b
1 holds not b
2 <> b
3 & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
3,b
2 & b
2,b
3 // b
4,b
4 & ( 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 ) & ( 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
4 // b
3,b
5 & b
3 <> b
5 ) ) & ( 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, b
4, b
5 being
Element of b
1 holds
not ( b
4,b
2 // b
2,b
5 & b
2 <> b
4 & ( for b
6 being
Element of b
1 holds
not ( b
3,b
2 // b
2,b
6 & b
3,b
4 // b
5,b
6 ) ) ) ) )
by Def7, REALSET2:def 7;
theorem Th48: :: DIRAF:48
theorem Th49: :: DIRAF:49
for b
1 being non
empty AffinStruct holds
( ( not for b
2, b
3 being
Element of b
1 holds not b
2 <> b
3 & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
3,b
2 & b
2,b
3 // b
4,b
4 & ( 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 ) & ( 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
4 // b
3,b
5 & b
3 <> b
5 ) ) & ( 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, b
4, b
5 being
Element of b
1 holds
not ( b
4,b
2 // b
2,b
5 & b
2 <> b
4 & ( for b
6 being
Element of b
1 holds
not ( b
3,b
2 // b
2,b
6 & b
3,b
4 // b
5,b
6 ) ) ) ) ) iff b
1 is
AffinSpace )
by Def7, REALSET2:def 7;
theorem Th50: :: DIRAF:50
theorem Th51: :: DIRAF:51
:: deftheorem Def8 defines 2-dimensional DIRAF:def 8 :
theorem Th52: :: DIRAF:52
canceled;
theorem Th53: :: DIRAF:53
theorem Th54: :: DIRAF:54
for b
1 being non
empty AffinStruct holds
( b
1 is
AffinPlane iff ( not for b
2, b
3 being
Element of b
1 holds not b
2 <> b
3 & ( for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
3,b
2 & b
2,b
3 // b
4,b
4 & ( 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 ) & ( 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
4 // b
3,b
5 & b
3 <> b
5 ) ) & ( 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, b
4, b
5 being
Element of b
1 holds
not ( b
4,b
2 // b
2,b
5 & b
2 <> b
4 & ( for b
6 being
Element of b
1 holds
not ( b
3,b
2 // b
2,b
6 & b
3,b
4 // b
5,b
6 ) ) ) ) & ( for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( not b
2,b
3 // b
4,b
5 & ( for b
6 being
Element of b
1 holds
not ( b
2,b
3 // b
2,b
6 & b
4,b
5 // b
4,b
6 ) ) ) ) ) )
by Def7, Def8, REALSET2:def 7;