:: PROJRED1 semantic presentation
theorem Th1: :: PROJRED1:1
theorem Th2: :: PROJRED1:2
theorem Th3: :: PROJRED1:3
for b
1 being
IncProjSpfor b
2, b
3 being
LINE of b
1 holds
not ( b
2 <> b
3 & ( for b
4, b
5 being
POINT of b
1 holds
not ( b
4 on b
2 & not b
4 on b
3 & b
5 on b
3 & not b
5 on b
2 ) ) )
theorem Th4: :: PROJRED1:4
for b
1 being
IncProjSpfor b
2, b
3 being
POINT of b
1 holds
not ( b
2 <> b
3 & ( for b
4, b
5 being
LINE of b
1 holds
not ( b
2 on b
4 & not b
2 on b
5 & b
3 on b
5 & not b
3 on b
4 ) ) )
theorem Th5: :: PROJRED1:5
for b
1 being
IncProjSpfor b
2 being
POINT of b
1 holds
ex b
3, b
4, b
5 being
LINE of b
1 st
( b
2 on b
3 & b
2 on b
4 & b
2 on b
5 & b
3 <> b
4 & b
4 <> b
5 & b
5 <> b
3 )
theorem Th6: :: PROJRED1:6
for b
1 being
IncProjSpfor b
2, b
3 being
LINE of b
1 holds
ex b
4 being
POINT of b
1 st
( not b
4 on b
2 & not b
4 on b
3 )
theorem Th7: :: PROJRED1:7
theorem Th8: :: PROJRED1:8
theorem Th9: :: PROJRED1:9
for b
1 being
IncProjSpfor b
2, b
3 being
POINT of b
1 holds
ex b
4 being
LINE of b
1 st
( not b
2 on b
4 & not b
3 on b
4 )
theorem Th10: :: PROJRED1:10
canceled;
theorem Th11: :: PROJRED1:11
canceled;
theorem Th12: :: PROJRED1:12
for b
1 being
IncProjSpfor b
2, b
3, b
4, b
5 being
POINT of b
1for b
6, b
7, b
8, b
9 being
LINE of b
1 holds
not ( b
2 on b
6 & b
2 on b
7 & b
6 <> b
7 & b
3 on b
6 & b
2 <> b
3 & b
4 on b
7 & b
5 on b
7 & b
4 <> b
5 & b
3 on b
8 & b
4 on b
8 & b
3 on b
9 & b
5 on b
9 & not b
8 <> b
9 )
theorem Th13: :: PROJRED1:13
for b
1 being
IncProjSpfor b
2, b
3, b
4 being
POINT of b
1for b
5 being
LINE of b
1 holds
( b
2,b
3,b
4 on b
5 implies ( b
2,b
4,b
3 on b
5 & b
3,b
2,b
4 on b
5 & b
3,b
4,b
2 on b
5 & b
4,b
2,b
3 on b
5 & b
4,b
3,b
2 on b
5 ) )
theorem Th14: :: PROJRED1:14
for b
1 being
Desarguesian IncProjSpfor 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
8 & b
2 <> b
3 & b
2 <> b
5 & b
6 <> b
5 & ( for b
21 being
LINE of b
1 holds
not b
9,b
10,b
11 on b
21 ) )
Lemma9:
for b1 being IncProjSp
for b2, b3, b4, b5 being POINT of b1
for b6, b7 being LINE of b1 holds
not ( ex b8 being POINT of b1 st
( b8 on b6 & b8 on b7 ) & b2 on b6 & b3 on b6 & b4 on b6 & b5 on b6 & b2,b3,b4,b5 are_mutually_different & ( for b8, b9, b10, b11 being POINT of b1 holds
not ( b8 on b7 & b9 on b7 & b10 on b7 & b11 on b7 & b8,b9,b10,b11 are_mutually_different ) ) )
theorem Th15: :: PROJRED1:15
for b
1 being
IncProjSp holds
( ex b
2 being
LINE of b
1ex b
3, b
4, b
5, b
6 being
POINT of b
1 st
( b
3 on b
2 & b
4 on b
2 & b
5 on b
2 & b
6 on b
2 & b
3,b
4,b
5,b
6 are_mutually_different ) implies for b
2 being
LINE of b
1 holds
ex b
3, b
4, b
5, b
6 being
POINT of b
1 st
( b
3 on b
2 & b
4 on b
2 & b
5 on b
2 & b
6 on b
2 & b
3,b
4,b
5,b
6 are_mutually_different ) )
Lemma11:
for b1 being Fanoian IncProjSp holds
ex b2, b3, b4, b5, b6, b7, b8 being POINT of b1ex b9, b10, b11, b12, b13, b14, b15, b16 being LINE of b1ex b17 being POINT of b1 st
( not b3 on b13 & not b4 on b13 & not b2 on b12 & not b5 on b12 & not b2 on b14 & not b4 on b14 & not b3 on b15 & not b5 on b15 & b6,b2,b5 on b13 & b6,b3,b4 on b12 & b7,b3,b5 on b14 & b7,b2,b4 on b15 & b8,b2,b3 on b9 & b8,b4,b5 on b10 & b6,b7 on b11 & not b8 on b11 & b7 on b16 & b8 on b16 & b11,b16,b14,b15 are_mutually_different & b17 on b9 & b8,b17,b2,b3 are_mutually_different )
theorem Th16: :: PROJRED1:16
for b
1 being
Fanoian IncProjSp holds
ex b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
POINT of b
1ex b
9, b
10, b
11, b
12, b
13, b
14, b
15, b
16 being
LINE of b
1 st
( not b
3 on b
13 & not b
4 on b
13 & not b
2 on b
12 & not b
5 on b
12 & not b
2 on b
14 & not b
4 on b
14 & not b
3 on b
15 & not b
5 on b
15 & b
6,b
2,b
5 on b
13 & b
6,b
3,b
4 on b
12 & b
7,b
3,b
5 on b
14 & b
7,b
2,b
4 on b
15 & b
8,b
2,b
3 on b
9 & b
8,b
4,b
5 on b
10 & b
6,b
7 on b
11 & not b
8 on b
11 )
theorem Th17: :: PROJRED1:17
theorem Th18: :: PROJRED1:18
theorem Th19: :: PROJRED1:19
definition
let c
1 be
2-dimensional Desarguesian IncProjSp;
let c
2, c
3 be
LINE of c
1;
let c
4 be
POINT of c
1;
assume E13:
( not c
4 on c
2 & not c
4 on c
3 )
;
func IncProj c
2,c
4,c
3 -> PartFunc of the
Points of a
1,the
Points of a
1 means :
Def1:
:: PROJRED1:def 1
(
dom a
5 c= the
Points of a
1 & ( for b
1 being
POINT of a
1 holds
( b
1 in dom a
5 iff b
1 on a
2 ) ) & ( for b
1, b
2 being
POINT of a
1 holds
( b
1 on a
2 & b
2 on a
3 implies ( a
5 . b
1 = b
2 iff ex b
3 being
LINE of a
1 st
( a
4 on b
3 & b
1 on b
3 & b
2 on b
3 ) ) ) ) );
existence
ex b1 being PartFunc of the Points of c1,the Points of c1 st
( dom b1 c= the Points of c1 & ( for b2 being POINT of c1 holds
( b2 in dom b1 iff b2 on c2 ) ) & ( for b2, b3 being POINT of c1 holds
( b2 on c2 & b3 on c3 implies ( b1 . b2 = b3 iff ex b4 being LINE of c1 st
( c4 on b4 & b2 on b4 & b3 on b4 ) ) ) ) )
uniqueness
for b1, b2 being PartFunc of the Points of c1,the Points of c1 holds
( dom b1 c= the Points of c1 & ( for b3 being POINT of c1 holds
( b3 in dom b1 iff b3 on c2 ) ) & ( for b3, b4 being POINT of c1 holds
( b3 on c2 & b4 on c3 implies ( b1 . b3 = b4 iff ex b5 being LINE of c1 st
( c4 on b5 & b3 on b5 & b4 on b5 ) ) ) ) & dom b2 c= the Points of c1 & ( for b3 being POINT of c1 holds
( b3 in dom b2 iff b3 on c2 ) ) & ( for b3, b4 being POINT of c1 holds
( b3 on c2 & b4 on c3 implies ( b2 . b3 = b4 iff ex b5 being LINE of c1 st
( c4 on b5 & b3 on b5 & b4 on b5 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines IncProj PROJRED1:def 1 :
theorem Th20: :: PROJRED1:20
canceled;
theorem Th21: :: PROJRED1:21
theorem Th22: :: PROJRED1:22
theorem Th23: :: PROJRED1:23
theorem Th24: :: PROJRED1:24
theorem Th25: :: PROJRED1:25
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3 being
POINT of b
1for b
4, b
5, b
6 being
LINE of b
1 holds
( not b
2 on b
4 & not b
2 on b
5 & not b
3 on b
5 & not b
3 on b
6 implies (
dom ((IncProj b5,b3,b6) * (IncProj b4,b2,b5)) = dom (IncProj b4,b2,b5) &
rng ((IncProj b5,b3,b6) * (IncProj b4,b2,b5)) = rng (IncProj b5,b3,b6) ) )
theorem Th26: :: PROJRED1:26
theorem Th27: :: PROJRED1:27
theorem Th28: :: PROJRED1:28
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3, b
4 being
POINT of b
1for b
5, b
6, b
7 being
LINE of b
1 holds
not ( not b
2 on b
5 & not b
2 on b
6 & not b
3 on b
6 & not b
3 on b
7 & b
4 on b
5 & b
4 on b
6 & b
4 on b
7 & b
5 <> b
7 & ( for b
8 being
POINT of b
1 holds
not ( not b
8 on b
5 & not b
8 on b
7 &
(IncProj b6,b3,b7) * (IncProj b5,b2,b6) = IncProj b
5,b
8,b
7 ) ) )