:: PROJRED2 semantic presentation
:: deftheorem Def1 defines are_concurrent PROJRED2:def 1 :
:: deftheorem Def2 defines CHAIN PROJRED2:def 2 :
:: deftheorem Def3 defines Projection PROJRED2:def 3 :
theorem Th1: :: PROJRED2:1
theorem Th2: :: PROJRED2:2
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3, b
4 being
LINE of b
1 holds
( b
2,b
3,b
4 are_concurrent implies ( b
2,b
4,b
3 are_concurrent & b
3,b
2,b
4 are_concurrent & b
3,b
4,b
2 are_concurrent & b
4,b
2,b
3 are_concurrent & b
4,b
3,b
2 are_concurrent ) )
theorem Th3: :: PROJRED2:3
theorem Th4: :: PROJRED2:4
canceled;
theorem Th5: :: PROJRED2:5
theorem Th6: :: PROJRED2:6
theorem Th7: :: PROJRED2:7
theorem Th8: :: PROJRED2:8
theorem Th9: :: PROJRED2:9
theorem Th10: :: PROJRED2:10
theorem Th11: :: PROJRED2:11
theorem Th12: :: PROJRED2:12
theorem Th13: :: PROJRED2:13
theorem Th14: :: PROJRED2:14
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 ( not b
2 on b
4 & not b
2 on b
5 & not b
3 on b
5 & not b
3 on b
6 & b
4,b
5,b
6 are_concurrent & b
4 <> b
6 & ( for b
7 being
POINT of b
1 holds
not ( not b
7 on b
4 & not b
7 on b
6 &
(IncProj b5,b3,b6) * (IncProj b4,b2,b5) = IncProj b
4,b
7,b
6 ) ) )
theorem Th15: :: PROJRED2:15
for b
1 being
2-dimensional Desarguesian IncProjSpfor 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, b
16 being
LINE of b
1 holds
( not b
2 on b
9 & not b
3 on b
10 & not b
2 on b
11 & not b
3 on b
11 & not b
9,b
10,b
11 are_concurrent & b
4 on b
9 & b
4 on b
11 & b
4 on b
12 & not b
3 on b
12 & b
9 <> b
12 & b
2 <> b
3 & b
3 <> b
5 & b
2 on b
13 & b
3 on b
13 & not b
10,b
11,b
13 are_concurrent & b
6 on b
11 & b
6 on b
10 & b
2 on b
14 & b
6 on b
14 & b
7 on b
9 & b
7 on b
14 & b
5 on b
13 & b
5 on b
15 & b
7 on b
15 & b
8 on b
15 & b
6 on b
16 & b
3 on b
16 & b
8 on b
16 & b
8 on b
12 & b
12 <> b
11 & b
5 <> b
2 & not b
5 on b
9 & not b
5 on b
12 implies
(IncProj b11,b3,b10) * (IncProj b9,b2,b11) = (IncProj b12,b3,b10) * (IncProj b9,b5,b12) )
theorem Th16: :: PROJRED2:16
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
POINT of b
1for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18 being
LINE of b
1 holds
( not b
2 on b
11 & not b
2 on b
12 & not b
3 on b
13 & not b
3 on b
12 & not b
3 on b
14 & not b
11,b
13,b
12 are_concurrent & b
2 <> b
3 & b
3 <> b
4 & b
11 <> b
14 & b
5,b
6 on b
11 & b
6,b
7,b
8 on b
13 & b
5,b
8,b
9 on b
12 & b
2,b
3,b
8 on b
15 & b
5,b
10 on b
14 & b
2,b
6,b
9 on b
16 & b
3,b
9,b
10 on b
17 & b
6,b
10,b
4 on b
18 & b
4 on b
15 implies
(IncProj b12,b3,b13) * (IncProj b11,b2,b12) = (IncProj b14,b3,b13) * (IncProj b11,b4,b14) )
theorem Th17: :: PROJRED2:17
for b
1 being
2-dimensional Desarguesian IncProjSpfor 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, b
16 being
LINE of b
1 holds
( not b
2 on b
9 & not b
2 on b
10 & not b
3 on b
11 & not b
3 on b
10 & not b
3 on b
12 & not b
9,b
11,b
10 are_concurrent & not b
11,b
10,b
13 are_concurrent & b
9 <> b
12 & b
12 <> b
10 & b
2 <> b
3 & b
4,b
5 on b
9 & b
6 on b
11 & b
4,b
6 on b
10 & b
2,b
3,b
7 on b
13 & b
4,b
8 on b
12 & b
2,b
6,b
5 on b
14 & b
7,b
5,b
8 on b
15 & b
3,b
6,b
8 on b
16 implies ( b
7 <> b
2 & b
7 <> b
3 & not b
7 on b
9 & not b
7 on b
12 ) )
theorem Th18: :: PROJRED2:18
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
POINT of b
1for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18 being
LINE of b
1 holds
( not b
2 on b
11 & not b
2 on b
12 & not b
3 on b
13 & not b
3 on b
12 & not b
3 on b
14 & not b
11,b
13,b
12 are_concurrent & b
2 <> b
3 & b
11 <> b
14 & b
4,b
5 on b
11 & b
5,b
6,b
7 on b
13 & b
4,b
7,b
8 on b
12 & b
2,b
3,b
7 on b
15 & b
4,b
9 on b
14 & b
2,b
5,b
8 on b
16 & b
3,b
8,b
9 on b
17 & b
5,b
9,b
10 on b
18 & b
10 on b
15 implies ( not b
10 on b
11 & not b
10 on b
14 & b
3 <> b
10 ) )
theorem Th19: :: PROJRED2:19
for b
1 being
2-dimensional Desarguesian IncProjSpfor 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, b
16 being
LINE of b
1 holds
( not b
2 on b
9 & not b
2 on b
10 & not b
3 on b
11 & not b
3 on b
10 & not b
4 on b
9 & not b
9,b
11,b
10 are_concurrent & not b
11,b
10,b
12 are_concurrent & b
2 <> b
3 & b
3 <> b
4 & b
4 <> b
2 & b
5,b
6 on b
9 & b
7 on b
11 & b
5,b
7 on b
10 & b
2,b
3,b
4 on b
12 & b
5,b
8 on b
13 & b
2,b
7,b
6 on b
14 & b
4,b
6,b
8 on b
15 & b
3,b
7,b
8 on b
16 implies ( b
13 <> b
9 & b
13 <> b
10 & not b
4 on b
13 & not b
3 on b
13 ) )
theorem Th20: :: PROJRED2:20
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
POINT of b
1for b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18 being
LINE of b
1 holds
( not b
2 on b
11 & not b
2 on b
12 & not b
3 on b
13 & not b
3 on b
12 & not b
4 on b
11 & not b
11,b
13,b
12 are_concurrent & b
2 <> b
3 & b
3 <> b
4 & b
5,b
6 on b
11 & b
6,b
7,b
8 on b
13 & b
5,b
8,b
9 on b
12 & b
2,b
3,b
8 on b
14 & b
5,b
10 on b
15 & b
2,b
6,b
9 on b
16 & b
3,b
9,b
10 on b
17 & b
6,b
10,b
4 on b
18 & b
4 on b
14 implies ( not b
3 on b
15 & not b
4 on b
15 & b
11 <> b
15 ) )
Lemma16:
for b1 being 2-dimensional Desarguesian IncProjSp
for b2, b3 being POINT of b1
for b4, b5, b6, b7, b8 being LINE of b1 holds
not ( not b2 on b4 & not b3 on b5 & not b2 on b6 & not b3 on b6 & not b4,b5,b6 are_concurrent & b4,b6,b7 are_concurrent & not b3 on b7 & b4 <> b7 & b2 <> b3 & b2 on b8 & b3 on b8 & not b5,b6,b8 are_concurrent & ( for b9 being POINT of b1 holds
not ( b9 on b8 & not b9 on b4 & not b9 on b7 & (IncProj b6,b3,b5) * (IncProj b4,b2,b6) = (IncProj b7,b3,b5) * (IncProj b4,b9,b7) ) ) )
Lemma17:
for b1 being 2-dimensional Desarguesian IncProjSp
for b2, b3 being POINT of b1
for b4, b5, b6, b7, b8 being LINE of b1 holds
not ( not b2 on b4 & not b3 on b5 & not b2 on b6 & not b3 on b6 & not b4,b5,b6 are_concurrent & b4,b6,b7 are_concurrent & not b3 on b7 & b4 <> b7 & b2 <> b3 & b2 on b8 & b3 on b8 & b5,b6,b8 are_concurrent & ( for b9 being POINT of b1 holds
not ( b9 on b8 & not b9 on b4 & not b9 on b7 & (IncProj b6,b3,b5) * (IncProj b4,b2,b6) = (IncProj b7,b3,b5) * (IncProj b4,b9,b7) ) ) )
theorem Th21: :: PROJRED2:21
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3 being
POINT of b
1for b
4, b
5, b
6, b
7, b
8 being
LINE of b
1 holds
not ( not b
2 on b
4 & not b
3 on b
5 & not b
2 on b
6 & not b
3 on b
6 & not b
4,b
5,b
6 are_concurrent & b
4,b
6,b
7 are_concurrent & not b
3 on b
7 & b
4 <> b
7 & b
2 <> b
3 & b
2 on b
8 & b
3 on b
8 & ( for b
9 being
POINT of b
1 holds
not ( b
9 on b
8 & not b
9 on b
4 & not b
9 on b
7 &
(IncProj b6,b3,b5) * (IncProj b4,b2,b6) = (IncProj b7,b3,b5) * (IncProj b4,b9,b7) ) ) )
theorem Th22: :: PROJRED2:22
for b
1 being
2-dimensional Desarguesian IncProjSpfor b
2, b
3 being
POINT of b
1for b
4, b
5, b
6, b
7, b
8 being
LINE of b
1 holds
not ( not b
2 on b
4 & not b
3 on b
5 & not b
2 on b
6 & not b
3 on b
6 & not b
4,b
5,b
6 are_concurrent & b
5,b
6,b
7 are_concurrent & not b
2 on b
7 & b
5 <> b
7 & b
2 <> b
3 & b
2 on b
8 & b
3 on b
8 & ( for b
9 being
POINT of b
1 holds
not ( b
9 on b
8 & not b
9 on b
5 & not b
9 on b
7 &
(IncProj b6,b3,b5) * (IncProj b4,b2,b6) = (IncProj b7,b9,b5) * (IncProj b4,b2,b7) ) ) )
theorem Th23: :: PROJRED2:23
for b
1 being
2-dimensional Desarguesian IncProjSpfor 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, b
12, b
13 being
LINE of b
1 holds
( not b
2 on b
8 & not b
3 on b
9 & not b
2 on b
10 & not b
3 on b
10 & not b
2 on b
9 & not b
3 on b
8 & b
4 on b
8 & b
4 on b
10 & b
5 on b
9 & b
5 on b
10 & b
2 on b
11 & b
5 on b
11 & b
4 on b
12 & b
3 on b
12 & b
6 on b
8 & b
6 on b
11 & b
7 on b
9 & b
7 on b
12 & b
6 on b
13 & b
7 on b
13 & not b
8,b
9,b
10 are_concurrent implies
(IncProj b10,b3,b9) * (IncProj b8,b2,b10) = (IncProj b13,b2,b9) * (IncProj b8,b3,b13) )
Lemma19:
for b1 being 2-dimensional Desarguesian IncProjSp
for b2, b3, b4, b5 being POINT of b1
for b6, b7, b8, b9 being LINE of b1 holds
not ( not b2 on b6 & not b3 on b7 & not b2 on b8 & not b3 on b8 & b4 on b6 & b4 on b8 & b2 <> b3 & b2 on b9 & b3 on b9 & b5 on b9 & not b5 on b6 & b5 <> b3 & not b6,b7,b8 are_concurrent & not b7,b8,b9 are_concurrent & ( for b10 being LINE of b1 holds
not ( b4 on b10 & not b3 on b10 & not b5 on b10 & (IncProj b8,b3,b7) * (IncProj b6,b2,b8) = (IncProj b10,b3,b7) * (IncProj b6,b5,b10) ) ) )
Lemma20:
for b1 being 2-dimensional Desarguesian IncProjSp
for b2, b3, b4, b5 being POINT of b1
for b6, b7, b8, b9 being LINE of b1 holds
not ( not b2 on b6 & not b3 on b7 & not b2 on b8 & not b3 on b8 & b4 on b6 & b4 on b8 & b2 <> b3 & b2 on b9 & b3 on b9 & b5 on b9 & not b5 on b6 & b5 <> b3 & not b6,b7,b8 are_concurrent & b7,b8,b9 are_concurrent & ( for b10 being LINE of b1 holds
not ( b4 on b10 & not b3 on b10 & not b5 on b10 & (IncProj b8,b3,b7) * (IncProj b6,b2,b8) = (IncProj b10,b3,b7) * (IncProj b6,b5,b10) ) ) )
theorem Th24: :: PROJRED2:24
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, b
8 being
LINE of b
1 holds
not ( not b
2 on b
5 & not b
3 on b
6 & not b
2 on b
7 & not b
3 on b
7 & b
2 <> b
3 & b
2 on b
8 & b
3 on b
8 & b
4 on b
8 & not b
4 on b
5 & b
4 <> b
3 & not b
5,b
6,b
7 are_concurrent & ( for b
9 being
LINE of b
1 holds
not ( b
5,b
7,b
9 are_concurrent & not b
3 on b
9 & not b
4 on b
9 &
(IncProj b7,b3,b6) * (IncProj b5,b2,b7) = (IncProj b9,b3,b6) * (IncProj b5,b4,b9) ) ) )
theorem Th25: :: PROJRED2:25
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, b
8 being
LINE of b
1 holds
not ( not b
2 on b
5 & not b
3 on b
6 & not b
2 on b
7 & not b
3 on b
7 & b
2 <> b
3 & b
2 on b
8 & b
3 on b
8 & b
4 on b
8 & not b
4 on b
6 & b
4 <> b
2 & not b
5,b
6,b
7 are_concurrent & ( for b
9 being
LINE of b
1 holds
not ( b
6,b
7,b
9 are_concurrent & not b
2 on b
9 & not b
4 on b
9 &
(IncProj b7,b3,b6) * (IncProj b5,b2,b7) = (IncProj b9,b4,b6) * (IncProj b5,b2,b9) ) ) )