:: INCPROJ semantic presentation
:: deftheorem Def1 defines ProjectiveLines INCPROJ:def 1 :
theorem Th1: :: INCPROJ:1
canceled;
theorem Th2: :: INCPROJ:2
definition
let c
1 be
proper CollSp;
func Proj_Inc c
1 -> Relation of the
carrier of a
1,
ProjectiveLines a
1 means :
Def2:
:: INCPROJ:def 2
for b
1, b
2 being
set holds
(
[b1,b2] in a
2 iff ( b
1 in the
carrier of a
1 & b
2 in ProjectiveLines a
1 & ex b
3 being
set st
( b
2 = b
3 & b
1 in b
3 ) ) );
existence
ex b1 being Relation of the carrier of c1, ProjectiveLines c1 st
for b2, b3 being set holds
( [b2,b3] in b1 iff ( b2 in the carrier of c1 & b3 in ProjectiveLines c1 & ex b4 being set st
( b3 = b4 & b2 in b4 ) ) )
uniqueness
for b1, b2 being Relation of the carrier of c1, ProjectiveLines c1 holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ( b3 in the carrier of c1 & b4 in ProjectiveLines c1 & ex b5 being set st
( b4 = b5 & b3 in b5 ) ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ( b3 in the carrier of c1 & b4 in ProjectiveLines c1 & ex b5 being set st
( b4 = b5 & b3 in b5 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Proj_Inc INCPROJ:def 2 :
:: deftheorem Def3 defines IncProjSp_of INCPROJ:def 3 :
theorem Th3: :: INCPROJ:3
canceled;
theorem Th4: :: INCPROJ:4
theorem Th5: :: INCPROJ:5
theorem Th6: :: INCPROJ:6
theorem Th7: :: INCPROJ:7
canceled;
theorem Th8: :: INCPROJ:8
theorem Th9: :: INCPROJ:9
theorem Th10: :: INCPROJ:10
theorem Th11: :: INCPROJ:11
theorem Th12: :: INCPROJ:12
theorem Th13: :: INCPROJ:13
theorem Th14: :: INCPROJ:14
theorem Th15: :: INCPROJ:15
theorem Th16: :: INCPROJ:16
theorem Th17: :: INCPROJ:17
theorem Th18: :: INCPROJ:18
theorem Th19: :: INCPROJ:19
theorem Th20: :: INCPROJ:20
for b
1 being
CollProjectiveSpace holds
( ( for b
2, b
3, b
4, b
5, b
6 being
Point of b
1 holds
ex b
7, b
8 being
Point of b
1 st
( b
2,b
4,b
7 is_collinear & b
3,b
5,b
8 is_collinear & b
6,b
7,b
8 is_collinear ) ) implies for b
2 being
POINT of
(IncProjSp_of b1)for b
3, b
4 being
LINE of
(IncProjSp_of b1) holds
ex b
5, b
6 being
POINT of
(IncProjSp_of b1)ex b
7 being
LINE of
(IncProjSp_of b1) st
( b
2 on b
7 & b
5 on b
7 & b
6 on b
7 & b
5 on b
3 & b
6 on b
4 ) )
:: deftheorem Def4 INCPROJ:def 4 :
canceled;
:: deftheorem Def5 defines are_mutually_different INCPROJ:def 5 :
:: deftheorem Def6 defines are_mutually_different INCPROJ:def 6 :
:: deftheorem Def7 defines on INCPROJ:def 7 :
for b
1 being
IncProjStr for b
2, b
3 being
POINT of b
1for b
4 being
LINE of b
1 holds
( b
2,b
3 on b
4 iff ( b
2 on b
4 & b
3 on b
4 ) );
:: deftheorem Def8 defines on INCPROJ:def 8 :
for b
1 being
IncProjStr for b
2, b
3 being
POINT of b
1for b
4 being
LINE of b
1for b
5 being
POINT of b
1 holds
( b
2,b
3,b
5 on b
4 iff ( b
2 on b
4 & b
3 on b
4 & b
5 on b
4 ) );
theorem Th21: :: INCPROJ:21
for b
1 being
CollProjectiveSpace holds
( ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Point of b
1 holds
not ( b
2,b
3,b
4 is_collinear & b
5,b
6,b
4 is_collinear & b
2,b
5,b
7 is_collinear & b
3,b
6,b
7 is_collinear & b
2,b
6,b
8 is_collinear & b
3,b
5,b
8 is_collinear & b
7,b
4,b
8 is_collinear & not b
2,b
3,b
6 is_collinear & not b
2,b
3,b
5 is_collinear & not b
2,b
5,b
6 is_collinear & not b
3,b
5,b
6 is_collinear ) ) implies for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
POINT of
(IncProjSp_of b1)for b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
LINE of
(IncProjSp_of b1) holds
not ( not b
3 on b
9 & not b
4 on b
9 & not b
2 on b
10 & not b
5 on b
10 & not b
2 on b
11 & not b
4 on b
11 & not b
3 on b
12 & not b
5 on b
12 & b
6,b
2,b
5 on b
9 & b
6,b
3,b
4 on b
10 & b
7,b
3,b
5 on b
11 & b
7,b
2,b
4 on b
12 & b
8,b
2,b
3 on b
13 & b
8,b
4,b
5 on b
14 & b
6,b
7 on b
15 & b
8 on b
15 ) )
theorem Th22: :: INCPROJ:22
for b
1 being
CollProjectiveSpace holds
( ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Point of b
1 holds
( b
2 <> b
6 & b
3 <> b
6 & b
2 <> b
7 & b
4 <> b
7 & b
2 <> b
8 & b
5 <> b
8 & not b
2,b
3,b
4 is_collinear & not b
2,b
3,b
5 is_collinear & not b
2,b
4,b
5 is_collinear & b
3,b
4,b
11 is_collinear & b
6,b
7,b
11 is_collinear & b
4,b
5,b
9 is_collinear & b
7,b
8,b
9 is_collinear & b
3,b
5,b
10 is_collinear & b
6,b
8,b
10 is_collinear & b
2,b
3,b
6 is_collinear & b
2,b
4,b
7 is_collinear & b
2,b
5,b
8 is_collinear implies b
9,b
10,b
11 is_collinear ) ) implies for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
POINT of
(IncProjSp_of b1)for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
LINE of
(IncProjSp_of b1) holds
not ( b
2,b
3,b
4 on b
12 & b
2,b
6,b
5 on b
13 & b
2,b
8,b
7 on b
14 & b
8,b
6,b
11 on b
15 & b
8,b
9,b
4 on b
16 & b
6,b
10,b
4 on b
17 & b
11,b
5,b
7 on b
18 & b
3,b
9,b
7 on b
19 & b
3,b
10,b
5 on b
20 & b
12,b
13,b
14 are_mutually_different & b
2 <> b
4 & b
2 <> b
6 & b
2 <> b
8 & b
2 <> b
3 & b
2 <> b
5 & b
2 <> b
7 & b
4 <> b
3 & b
6 <> b
5 & b
8 <> b
7 & ( for b
21 being
LINE of
(IncProjSp_of b1) holds
not b
9,b
10,b
11 on b
21 ) ) )
theorem Th23: :: INCPROJ:23
for b
1 being
CollProjectiveSpace holds
( ( for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Point of b
1 holds
( b
2 <> b
4 & b
2 <> b
5 & b
4 <> b
5 & b
3 <> b
4 & b
3 <> b
5 & b
2 <> b
7 & b
2 <> b
8 & b
7 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & not b
2,b
3,b
6 is_collinear & b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
6,b
7 is_collinear & b
2,b
6,b
8 is_collinear & b
3,b
7,b
11 is_collinear & b
6,b
4,b
11 is_collinear & b
3,b
8,b
10 is_collinear & b
5,b
6,b
10 is_collinear & b
4,b
8,b
9 is_collinear & b
5,b
7,b
9 is_collinear implies b
9,b
10,b
11 is_collinear ) ) implies for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
POINT of
(IncProjSp_of b1)for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
LINE of
(IncProjSp_of b1) holds
( b
2,b
3,b
4,b
5 are_mutually_different & b
2,b
6,b
7,b
8 are_mutually_different & b
14 <> b
17 & b
2 on b
14 & b
2 on b
17 & b
4,b
8,b
9 on b
12 & b
5,b
6,b
10 on b
15 & b
3,b
7,b
11 on b
18 & b
3,b
8,b
10 on b
13 & b
5,b
7,b
9 on b
16 & b
4,b
6,b
11 on b
19 & b
6,b
7,b
8 on b
14 & b
3,b
4,b
5 on b
17 & b
9,b
10 on b
20 implies b
11 on b
20 ) )
definition
let c
1 be
IncProjStr ;
attr a
1 is
partial means :
Def9:
:: INCPROJ:def 9
for b
1, b
2 being
POINT of a
1for b
3, b
4 being
LINE of a
1 holds
not ( b
1 on b
3 & b
2 on b
3 & b
1 on b
4 & b
2 on b
4 & not b
1 = b
2 & not b
3 = b
4 );
attr a
1 is
linear means :
Def10:
:: INCPROJ:def 10
for b
1, b
2 being
POINT of a
1 holds
ex b
3 being
LINE of a
1 st
( b
1 on b
3 & b
2 on b
3 );
attr a
1 is
up-2-dimensional means :
Def11:
:: INCPROJ:def 11
not for b
1 being
POINT of a
1for b
2 being
LINE of a
1 holds b
1 on b
2;
attr a
1 is
up-3-rank means :
Def12:
:: INCPROJ:def 12
for b
1 being
LINE of a
1 holds
ex b
2, b
3, b
4 being
POINT of a
1 st
( b
2 <> b
3 & b
3 <> b
4 & b
4 <> b
2 & b
2 on b
1 & b
3 on b
1 & b
4 on b
1 );
attr a
1 is
Vebleian means :
Def13:
:: INCPROJ:def 13
for b
1, b
2, b
3, b
4, b
5, b
6 being
POINT of a
1for b
7, b
8, b
9, b
10 being
LINE of a
1 holds
not ( b
1 on b
7 & b
2 on b
7 & b
3 on b
8 & b
4 on b
8 & b
5 on b
7 & b
5 on b
8 & b
1 on b
9 & b
3 on b
9 & b
2 on b
10 & b
4 on b
10 & not b
5 on b
9 & not b
5 on b
10 & b
7 <> b
8 & ( for b
11 being
POINT of a
1 holds
not ( b
11 on b
9 & b
11 on b
10 ) ) );
end;
:: deftheorem Def9 defines partial INCPROJ:def 9 :
for b
1 being
IncProjStr holds
( b
1 is
partial iff for b
2, b
3 being
POINT of b
1for b
4, b
5 being
LINE of b
1 holds
not ( b
2 on b
4 & b
3 on b
4 & b
2 on b
5 & b
3 on b
5 & not b
2 = b
3 & not b
4 = b
5 ) );
:: deftheorem Def10 defines linear INCPROJ:def 10 :
:: deftheorem Def11 defines up-2-dimensional INCPROJ:def 11 :
:: deftheorem Def12 defines up-3-rank INCPROJ:def 12 :
:: deftheorem Def13 defines Vebleian INCPROJ:def 13 :
for b
1 being
IncProjStr holds
( b
1 is
Vebleian iff for b
2, b
3, b
4, b
5, b
6, b
7 being
POINT of b
1for b
8, b
9, b
10, b
11 being
LINE of b
1 holds
not ( b
2 on b
8 & b
3 on b
8 & b
4 on b
9 & b
5 on b
9 & b
6 on b
8 & b
6 on b
9 & b
2 on b
10 & b
4 on b
10 & b
3 on b
11 & b
5 on b
11 & not b
6 on b
10 & not b
6 on b
11 & b
8 <> b
9 & ( for b
12 being
POINT of b
1 holds
not ( b
12 on b
10 & b
12 on b
11 ) ) ) );
:: deftheorem Def14 defines 2-dimensional INCPROJ:def 14 :
:: deftheorem Def15 INCPROJ:def 15 :
canceled;
:: deftheorem Def16 defines at_most-3-dimensional INCPROJ:def 16 :
:: deftheorem Def17 defines 3-dimensional INCPROJ:def 17 :
definition
let c
1 be
IncProjSp;
attr a
1 is
Fanoian means :
Def18:
:: INCPROJ:def 18
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
POINT of a
1for b
8, b
9, b
10, b
11, b
12, b
13, b
14 being
LINE of a
1 holds
not ( not b
2 on b
8 & not b
3 on b
8 & not b
1 on b
9 & not b
4 on b
9 & not b
1 on b
10 & not b
3 on b
10 & not b
2 on b
11 & not b
4 on b
11 & b
5,b
1,b
4 on b
8 & b
5,b
2,b
3 on b
9 & b
6,b
2,b
4 on b
10 & b
6,b
1,b
3 on b
11 & b
7,b
1,b
2 on b
12 & b
7,b
3,b
4 on b
13 & b
5,b
6 on b
14 & b
7 on b
14 );
end;
:: deftheorem Def18 defines Fanoian INCPROJ:def 18 :
for b
1 being
IncProjSp holds
( b
1 is
Fanoian iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
POINT of b
1for b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
LINE of b
1 holds
not ( not b
3 on b
9 & not b
4 on b
9 & not b
2 on b
10 & not b
5 on b
10 & not b
2 on b
11 & not b
4 on b
11 & not b
3 on b
12 & not b
5 on b
12 & b
6,b
2,b
5 on b
9 & b
6,b
3,b
4 on b
10 & b
7,b
3,b
5 on b
11 & b
7,b
2,b
4 on b
12 & b
8,b
2,b
3 on b
13 & b
8,b
4,b
5 on b
14 & b
6,b
7 on b
15 & b
8 on b
15 ) );
definition
let c
1 be
IncProjSp;
attr a
1 is
Desarguesian means :
Def19:
:: INCPROJ:def 19
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
POINT of a
1for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
LINE of a
1 holds
not ( b
1,b
2,b
3 on b
11 & b
1,b
5,b
4 on b
12 & b
1,b
7,b
6 on b
13 & b
7,b
5,b
10 on b
14 & b
7,b
8,b
3 on b
15 & b
5,b
9,b
3 on b
16 & b
10,b
4,b
6 on b
17 & b
2,b
8,b
6 on b
18 & b
2,b
9,b
4 on b
19 & b
11,b
12,b
13 are_mutually_different & b
1 <> b
3 & b
1 <> b
5 & b
1 <> b
7 & b
1 <> b
2 & b
1 <> b
4 & b
1 <> b
6 & b
3 <> b
2 & b
5 <> b
4 & b
7 <> b
6 & ( for b
20 being
LINE of a
1 holds
not b
8,b
9,b
10 on b
20 ) );
end;
:: deftheorem Def19 defines Desarguesian INCPROJ:def 19 :
for b
1 being
IncProjSp holds
( b
1 is
Desarguesian iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
POINT of b
1for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
LINE of b
1 holds
not ( b
2,b
3,b
4 on b
12 & b
2,b
6,b
5 on b
13 & b
2,b
8,b
7 on b
14 & b
8,b
6,b
11 on b
15 & b
8,b
9,b
4 on b
16 & b
6,b
10,b
4 on b
17 & b
11,b
5,b
7 on b
18 & b
3,b
9,b
7 on b
19 & b
3,b
10,b
5 on b
20 & b
12,b
13,b
14 are_mutually_different & b
2 <> b
4 & b
2 <> b
6 & b
2 <> b
8 & b
2 <> b
3 & b
2 <> b
5 & b
2 <> b
7 & b
4 <> b
3 & b
6 <> b
5 & b
8 <> b
7 & ( for b
21 being
LINE of b
1 holds
not b
9,b
10,b
11 on b
21 ) ) );
definition
let c
1 be
IncProjSp;
attr a
1 is
Pappian means :
Def20:
:: INCPROJ:def 20
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
POINT of a
1for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
LINE of a
1 holds
( b
1,b
2,b
3,b
4 are_mutually_different & b
1,b
5,b
6,b
7 are_mutually_different & b
13 <> b
16 & b
1 on b
13 & b
1 on b
16 & b
3,b
7,b
8 on b
11 & b
4,b
5,b
9 on b
14 & b
2,b
6,b
10 on b
17 & b
2,b
7,b
9 on b
12 & b
4,b
6,b
8 on b
15 & b
3,b
5,b
10 on b
18 & b
5,b
6,b
7 on b
13 & b
2,b
3,b
4 on b
16 & b
8,b
9 on b
19 implies b
10 on b
19 );
end;
:: deftheorem Def20 defines Pappian INCPROJ:def 20 :
for b
1 being
IncProjSp holds
( b
1 is
Pappian iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
POINT of b
1for b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20 being
LINE of b
1 holds
( b
2,b
3,b
4,b
5 are_mutually_different & b
2,b
6,b
7,b
8 are_mutually_different & b
14 <> b
17 & b
2 on b
14 & b
2 on b
17 & b
4,b
8,b
9 on b
12 & b
5,b
6,b
10 on b
15 & b
3,b
7,b
11 on b
18 & b
3,b
8,b
10 on b
13 & b
5,b
7,b
9 on b
16 & b
4,b
6,b
11 on b
19 & b
6,b
7,b
8 on b
14 & b
3,b
4,b
5 on b
17 & b
9,b
10 on b
20 implies b
11 on b
20 ) );