:: PROJPL_1 semantic presentation
:: deftheorem Def1 defines |' PROJPL_1:def 1 :
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 |' b
4 iff ( b
2 |' b
4 & b
3 |' b
4 ) );
:: deftheorem Def2 defines on PROJPL_1:def 2 :
for b
1 being
IncProjStr for b
2 being
POINT of b
1for b
3, b
4 being
LINE of b
1 holds
( b
2 on b
3,b
4 iff ( b
2 on b
3 & b
2 on b
4 ) );
:: deftheorem Def3 defines on PROJPL_1:def 3 :
for b
1 being
IncProjStr for b
2 being
POINT of b
1for b
3, b
4, b
5 being
LINE of b
1 holds
( b
2 on b
3,b
4,b
5 iff ( b
2 on b
3 & b
2 on b
4 & b
2 on b
5 ) );
theorem Th1: :: PROJPL_1:1
for b
1 being
IncProjStr for b
2, b
3, b
4 being
POINT of b
1for b
5, b
6, b
7 being
LINE of b
1 holds
( ( b
2,b
3 on b
5 implies b
3,b
2 on b
5 ) & ( 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 ) ) & ( b
2 on b
5,b
6 implies b
2 on b
6,b
5 ) & ( b
2 on b
5,b
6,b
7 implies ( b
2 on b
5,b
7,b
6 & b
2 on b
6,b
5,b
7 & b
2 on b
6,b
7,b
5 & b
2 on b
7,b
5,b
6 & b
2 on b
7,b
6,b
5 ) ) )
:: deftheorem Def4 defines configuration PROJPL_1:def 4 :
theorem Th2: :: PROJPL_1:2
theorem Th3: :: PROJPL_1:3
theorem Th4: :: PROJPL_1:4
for b
1 being
IncProjStr holds
( b
1 is
IncProjSp iff ( b
1 is
configuration & ( for b
2, b
3 being
POINT of b
1 holds
ex b
4 being
LINE of b
1 st b
2,b
3 on b
4 ) & not for b
2 being
POINT of b
1for b
3 being
LINE of b
1 holds not b
2 |' b
3 & ( for b
2 being
LINE of b
1 holds
ex b
3, b
4, b
5 being
POINT of b
1 st
( b
3,b
4,b
5 are_mutually_different & b
3,b
4,b
5 on b
2 ) ) & ( for b
2, b
3, b
4, b
5, b
6 being
POINT of b
1for b
7, b
8, b
9, b
10 being
LINE of b
1 holds
not ( b
2,b
3,b
6 on b
7 & b
4,b
5,b
6 on b
8 & b
2,b
4 on b
9 & b
3,b
5 on b
10 & b
6 |' b
9 & b
6 |' b
10 & b
7 <> b
8 & ( for b
11 being
POINT of b
1 holds
not b
11 on b
9,b
10 ) ) ) ) )
:: deftheorem Def5 defines is_collinear PROJPL_1:def 5 :
theorem Th5: :: PROJPL_1:5
theorem Th6: :: PROJPL_1:6
definition
let c
1 be
IncProjStr ;
let c
2, c
3, c
4, c
5 be
POINT of c
1;
pred c
2,c
3,c
4,c
5 is_a_quadrangle means :
Def6:
:: PROJPL_1:def 6
( a
2,a
3,a
4 is_a_triangle & a
3,a
4,a
5 is_a_triangle & a
4,a
5,a
2 is_a_triangle & a
5,a
2,a
3 is_a_triangle );
end;
:: deftheorem Def6 defines is_a_quadrangle PROJPL_1:def 6 :
for b
1 being
IncProjStr for b
2, b
3, b
4, b
5 being
POINT of b
1 holds
( b
2,b
3,b
4,b
5 is_a_quadrangle iff ( b
2,b
3,b
4 is_a_triangle & b
3,b
4,b
5 is_a_triangle & b
4,b
5,b
2 is_a_triangle & b
5,b
2,b
3 is_a_triangle ) );
theorem Th7: :: PROJPL_1:7
theorem Th8: :: PROJPL_1:8
theorem Th9: :: PROJPL_1:9
theorem Th10: :: PROJPL_1:10
theorem Th11: :: PROJPL_1:11
for b
1 being
IncProjStr for b
2, b
3, b
4 being
POINT 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 Th12: :: PROJPL_1:12
for b
1 being
IncProjStr for b
2, b
3, b
4 being
POINT of b
1 holds
( b
2,b
3,b
4 is_a_triangle implies ( b
2,b
4,b
3 is_a_triangle & b
3,b
2,b
4 is_a_triangle & b
3,b
4,b
2 is_a_triangle & b
4,b
2,b
3 is_a_triangle & b
4,b
3,b
2 is_a_triangle ) )
by Th11;
theorem Th13: :: PROJPL_1:13
for b
1 being
IncProjStr for b
2, b
3, b
4, b
5 being
POINT of b
1 holds
( b
2,b
3,b
4,b
5 is_a_quadrangle implies ( b
2,b
4,b
3,b
5 is_a_quadrangle & b
3,b
2,b
4,b
5 is_a_quadrangle & b
3,b
4,b
2,b
5 is_a_quadrangle & b
4,b
2,b
3,b
5 is_a_quadrangle & b
4,b
3,b
2,b
5 is_a_quadrangle & b
2,b
3,b
5,b
4 is_a_quadrangle & b
2,b
4,b
5,b
3 is_a_quadrangle & b
3,b
2,b
5,b
4 is_a_quadrangle & b
3,b
4,b
5,b
2 is_a_quadrangle & b
4,b
2,b
5,b
3 is_a_quadrangle & b
4,b
3,b
5,b
2 is_a_quadrangle & b
2,b
5,b
3,b
4 is_a_quadrangle & b
2,b
5,b
4,b
3 is_a_quadrangle & b
3,b
5,b
2,b
4 is_a_quadrangle & b
3,b
5,b
4,b
2 is_a_quadrangle & b
4,b
5,b
2,b
3 is_a_quadrangle & b
4,b
5,b
3,b
2 is_a_quadrangle & b
5,b
2,b
3,b
4 is_a_quadrangle & b
5,b
2,b
4,b
3 is_a_quadrangle & b
5,b
3,b
2,b
4 is_a_quadrangle & b
5,b
3,b
4,b
2 is_a_quadrangle & b
5,b
4,b
2,b
3 is_a_quadrangle & b
5,b
4,b
3,b
2 is_a_quadrangle ) )
theorem Th14: :: PROJPL_1:14
for b
1 being
IncProjStr for b
2, b
3, b
4, b
5 being
POINT of b
1for b
6, b
7 being
LINE of b
1 holds
( b
1 is
configuration & b
2,b
3 on b
6 & b
4,b
5 on b
7 & b
2,b
3 |' b
7 & b
4,b
5 |' b
6 & b
2 <> b
3 & b
4 <> b
5 implies b
2,b
3,b
4,b
5 is_a_quadrangle )
theorem Th15: :: PROJPL_1:15
definition
let c
1 be
IncProjSp;
mode Quadrangle of c
1 -> Element of
[:the Points of a1,the Points of a1,the Points of a1,the Points of a1:] means :: PROJPL_1:def 7
a
2 `1 ,a
2 `2 ,a
2 `3 ,a
2 `4 is_a_quadrangle ;
existence
ex b1 being Element of [:the Points of c1,the Points of c1,the Points of c1,the Points of c1:] st b1 `1 ,b1 `2 ,b1 `3 ,b1 `4 is_a_quadrangle
end;
:: deftheorem Def7 defines Quadrangle PROJPL_1:def 7 :
:: deftheorem Def8 defines * PROJPL_1:def 8 :
for b
1 being
IncProjSpfor b
2, b
3 being
POINT of b
1 holds
( b
2 <> b
3 implies for b
4 being
LINE of b
1 holds
( b
4 = b
2 * b
3 iff b
2,b
3 on b
4 ) );
theorem Th16: :: PROJPL_1:16
for b
1 being
IncProjSpfor b
2, b
3 being
POINT of b
1for b
4 being
LINE of b
1 holds
( b
2 <> b
3 implies ( b
2 on b
2 * b
3 & b
3 on b
2 * b
3 & b
2 * b
3 = b
3 * b
2 & ( b
2 on b
4 & b
3 on b
4 implies b
4 = b
2 * b
3 ) ) )
theorem Th17: :: PROJPL_1:17
for b
1 being
IncProjStr holds
not ( not for b
2, b
3, b
4 being
POINT of b
1 holds not b
2,b
3,b
4 is_a_triangle & ( for b
2, b
3 being
POINT of b
1 holds
ex b
4 being
LINE of b
1 st b
2,b
3 on b
4 ) & ( for b
2 being
POINT of b
1for b
3 being
LINE of b
1 holds not b
2 |' b
3 ) )
theorem Th18: :: PROJPL_1:18
for b
1 being
IncProjStr holds
not ( ex b
2, b
3, b
4, b
5 being
POINT of b
1 st b
2,b
3,b
4,b
5 is_a_quadrangle & ( for b
2, b
3, b
4 being
POINT of b
1 holds not b
2,b
3,b
4 is_a_triangle ) )
theorem Th19: :: PROJPL_1:19
theorem Th20: :: PROJPL_1:20
for b
1 being
IncProjStr for b
2, b
3, b
4, b
5 being
POINT of b
1for b
6, b
7, b
8 being
LINE of b
1 holds
( b
2,b
3,b
4,b
5 is_a_quadrangle & b
2,b
3 on b
6 & b
2,b
4 on b
7 & b
2,b
5 on b
8 implies b
6,b
7,b
8 are_mutually_different )
theorem Th21: :: PROJPL_1:21
for b
1 being
IncProjStr for 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
( b
1 is
configuration & b
2 on b
6,b
7,b
8 & b
6,b
7,b
8 are_mutually_different & b
2 |' b
9 & b
3 on b
9,b
6 & b
4 on b
9,b
7 & b
5 on b
9,b
8 implies b
3,b
4,b
5 are_mutually_different )
theorem Th22: :: PROJPL_1:22
for b
1 being
IncProjStr holds
( b
1 is
configuration & ( for b
2, b
3 being
POINT of b
1 holds
ex b
4 being
LINE of b
1 st b
2,b
3 on b
4 ) & ( for b
2, b
3 being
LINE of b
1 holds
ex b
4 being
POINT of b
1 st b
4 on b
2,b
3 ) & ex b
2, b
3, b
4, b
5 being
POINT of b
1 st b
2,b
3,b
4,b
5 is_a_quadrangle implies for b
2 being
LINE of b
1 holds
ex b
3, b
4, b
5 being
POINT of b
1 st
( b
3,b
4,b
5 are_mutually_different & b
3,b
4,b
5 on b
2 ) )
theorem Th23: :: PROJPL_1:23
for b
1 being
IncProjStr holds
( b
1 is
IncProjectivePlane iff ( b
1 is
configuration & ( for b
2, b
3 being
POINT of b
1 holds
ex b
4 being
LINE of b
1 st b
2,b
3 on b
4 ) & ( for b
2, b
3 being
LINE of b
1 holds
ex b
4 being
POINT of b
1 st b
4 on b
2,b
3 ) & ex b
2, b
3, b
4, b
5 being
POINT of b
1 st b
2,b
3,b
4,b
5 is_a_quadrangle ) )
:: deftheorem Def9 defines * PROJPL_1:def 9 :
theorem Th24: :: PROJPL_1:24
theorem Th25: :: PROJPL_1:25
theorem Th26: :: PROJPL_1:26
theorem Th27: :: PROJPL_1:27
theorem Th28: :: PROJPL_1:28
for b
1 being
IncProjectivePlanefor b
2, b
3, b
4, b
5 being
POINT of b
1 holds
not ( b
2 * b
4 = b
3 * b
5 & not b
2 = b
4 & not b
3 = b
5 & not b
4 = b
5 & not b
2 * b
4 = b
4 * b
5 )
theorem Th29: :: PROJPL_1:29
for b
1 being
IncProjectivePlanefor b
2, b
3, b
4, b
5 being
POINT of b
1 holds
not ( b
2 * b
4 = b
3 * b
5 & not b
2 = b
4 & not b
3 = b
5 & not b
4 = b
5 & not b
2 on b
4 * b
5 )
theorem Th30: :: PROJPL_1:30
theorem Th31: :: PROJPL_1:31
theorem Th32: :: PROJPL_1:32
for b
1 being
IncProjSpfor b
2, b
3, b
4, b
5 being
POINT of b
1 holds
( b
4 on b
2 * b
3 & b
4 on b
2 * b
5 & b
4 <> b
2 & b
5 <> b
2 & b
2 <> b
3 implies b
5 on b
2 * b
3 )
theorem Th33: :: PROJPL_1:33
for b
1 being
IncProjSpfor b
2, b
3, b
4 being
POINT of b
1 holds
( b
4 on b
2 * b
3 & b
2 <> b
4 implies b
3 on b
2 * b
4 )
theorem Th34: :: PROJPL_1:34
theorem Th35: :: PROJPL_1:35
theorem Th36: :: PROJPL_1:36
theorem Th37: :: PROJPL_1:37
for b
1 being
IncProjectivePlanefor b
2, b
3, b
4, b
5, b
6 being
POINT of b
1for b
7, b
8 being
LINE of b
1 holds
not ( b
3 on b
8 & b
4 on b
8 & b
2 on b
7 & b
2 |' b
8 & b
3 <> b
4 & b
5 <> b
2 & b
6 <> b
2 & b
5 on b
2 * b
3 & b
6 on b
2 * b
4 & ( for b
9 being
POINT of b
1 holds
not ( b
9 on b
5 * b
6 & b
9 on b
7 & b
9 <> b
2 ) ) )