:: PROJDES1 semantic presentation
theorem Th1: :: PROJDES1:1
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3,b
4 is_collinear implies ( b
3,b
4,b
2 is_collinear & b
4,b
2,b
3 is_collinear & b
3,b
2,b
4 is_collinear & b
2,b
4,b
3 is_collinear & b
4,b
3,b
2 is_collinear ) )
theorem Th2: :: PROJDES1:2
canceled;
theorem Th3: :: PROJDES1:3
canceled;
theorem Th4: :: PROJDES1:4
canceled;
theorem Th5: :: PROJDES1:5
theorem Th6: :: PROJDES1:6
theorem Th7: :: PROJDES1:7
theorem Th8: :: PROJDES1:8
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
not ( not b
2,b
3,b
4 is_collinear & b
2,b
4,b
5 is_collinear & b
3,b
4,b
6 is_collinear & b
4 <> b
5 & b
7,b
5,b
6 is_collinear & b
2,b
3,b
7 is_collinear & b
2 <> b
7 & not b
6 <> b
4 )
Lemma3:
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b6 is_collinear & b2 <> b5 & not b5 <> b6 )
definition
let c
1 be
up-3-dimensional CollProjectiveSpace;
let c
2, c
3, c
4, c
5 be
Element of c
1;
pred c
2,c
3,c
4,c
5 are_coplanar means :
Def1:
:: PROJDES1:def 1
ex b
1 being
Element of a
1 st
( a
2,a
3,b
1 is_collinear & a
4,a
5,b
1 is_collinear );
end;
:: deftheorem Def1 defines are_coplanar PROJDES1:def 1 :
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3,b
4,b
5 are_coplanar iff ex b
6 being
Element of b
1 st
( b
2,b
3,b
6 is_collinear & b
4,b
5,b
6 is_collinear ) );
theorem Th9: :: PROJDES1:9
canceled;
theorem Th10: :: PROJDES1:10
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( not ( not b
2,b
3,b
4 is_collinear & not b
3,b
4,b
5 is_collinear & not b
4,b
5,b
2 is_collinear & not b
5,b
2,b
3 is_collinear ) implies b
2,b
3,b
4,b
5 are_coplanar )
Lemma6:
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3,b4,b5 are_coplanar implies b3,b2,b4,b5 are_coplanar )
Lemma7:
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3,b4,b5 are_coplanar implies b3,b4,b5,b2 are_coplanar )
theorem Th11: :: PROJDES1:11
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3,b
4,b
5 are_coplanar implies ( b
3,b
4,b
5,b
2 are_coplanar & b
4,b
5,b
2,b
3 are_coplanar & b
5,b
2,b
3,b
4 are_coplanar & b
3,b
2,b
4,b
5 are_coplanar & b
4,b
3,b
5,b
2 are_coplanar & b
5,b
4,b
2,b
3 are_coplanar & b
2,b
5,b
3,b
4 are_coplanar & b
2,b
4,b
5,b
3 are_coplanar & b
3,b
5,b
2,b
4 are_coplanar & b
4,b
2,b
3,b
5 are_coplanar & b
5,b
3,b
4,b
2 are_coplanar & b
4,b
2,b
5,b
3 are_coplanar & b
5,b
3,b
2,b
4 are_coplanar & b
2,b
4,b
3,b
5 are_coplanar & b
3,b
5,b
4,b
2 are_coplanar & b
2,b
3,b
5,b
4 are_coplanar & b
2,b
5,b
4,b
3 are_coplanar & b
3,b
4,b
2,b
5 are_coplanar & b
3,b
2,b
5,b
4 are_coplanar & b
4,b
3,b
2,b
5 are_coplanar & b
4,b
5,b
3,b
2 are_coplanar & b
5,b
2,b
4,b
3 are_coplanar & b
5,b
4,b
3,b
2 are_coplanar ) )
Lemma9:
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( not b2,b3,b4 is_collinear & b2,b3,b4,b5 are_coplanar & b2,b3,b4,b6 are_coplanar implies b2,b3,b5,b6 are_coplanar )
theorem Th12: :: PROJDES1:12
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( not b
2,b
3,b
4 is_collinear & b
2,b
3,b
4,b
5 are_coplanar & b
2,b
3,b
4,b
6 are_coplanar & b
2,b
3,b
4,b
7 are_coplanar & b
2,b
3,b
4,b
8 are_coplanar implies b
5,b
6,b
7,b
8 are_coplanar )
Lemma11:
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( not b2,b3,b4 is_collinear & b2,b3,b4,b5 are_coplanar & b2,b3,b4,b6 are_coplanar & b2,b3,b4,b7 are_coplanar implies b2,b5,b6,b7 are_coplanar )
theorem Th13: :: PROJDES1:13
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( not b
2,b
3,b
4 is_collinear & b
5,b
6,b
7,b
2 are_coplanar & b
5,b
6,b
7,b
4 are_coplanar & b
5,b
6,b
7,b
3 are_coplanar & b
2,b
3,b
4,b
8 are_coplanar implies b
5,b
6,b
7,b
8 are_coplanar )
Lemma12:
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
( b2 <> b3 & b2,b3,b4 is_collinear & b5,b6,b2,b3 are_coplanar implies b5,b6,b2,b4 are_coplanar )
theorem Th14: :: PROJDES1:14
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & b
2,b
3,b
4 is_collinear & b
5,b
6,b
7,b
2 are_coplanar & b
5,b
6,b
7,b
3 are_coplanar implies b
5,b
6,b
7,b
4 are_coplanar )
theorem Th15: :: PROJDES1:15
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
not ( not b
2,b
3,b
4 is_collinear & b
2,b
3,b
4,b
5 are_coplanar & b
2,b
3,b
4,b
6 are_coplanar & b
2,b
3,b
4,b
7 are_coplanar & b
2,b
3,b
4,b
8 are_coplanar & ( for b
9 being
Element of b
1 holds
not ( b
5,b
6,b
9 is_collinear & b
7,b
8,b
9 is_collinear ) ) )
theorem Th16: :: PROJDES1:16
theorem Th17: :: PROJDES1:17
theorem Th18: :: PROJDES1:18
theorem Th19: :: PROJDES1:19
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( not b
2,b
3,b
4,b
5 are_coplanar & b
5,b
2,b
6 is_collinear & b
2 <> b
6 & b
2,b
3,b
4,b
6 are_coplanar )
theorem Th20: :: PROJDES1:20
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
( not b
2,b
3,b
4 is_collinear & not b
5,b
6,b
7 is_collinear & b
2,b
3,b
4,b
8 are_coplanar & b
2,b
3,b
4,b
9 are_coplanar & b
2,b
3,b
4,b
10 are_coplanar & b
5,b
6,b
7,b
8 are_coplanar & b
5,b
6,b
7,b
9 are_coplanar & b
5,b
6,b
7,b
10 are_coplanar & not b
2,b
3,b
4,b
5 are_coplanar implies b
8,b
9,b
10 is_collinear )
theorem Th21: :: PROJDES1:21
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( b
2 <> b
3 & b
4,b
2,b
3 is_collinear & not b
2,b
5,b
6,b
4 are_coplanar & not b
3,b
7,b
8 is_collinear & b
2,b
5,b
9 is_collinear & b
3,b
7,b
9 is_collinear & b
5,b
6,b
10 is_collinear & b
7,b
8,b
10 is_collinear & b
2,b
6,b
11 is_collinear & b
3,b
8,b
11 is_collinear implies b
9,b
10,b
11 is_collinear )
theorem Th22: :: PROJDES1:22
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( not b
2,b
3,b
4,b
5 are_coplanar & b
2,b
3,b
4,b
6 are_coplanar & not b
2,b
3,b
6 is_collinear & b
2,b
3,b
5,b
6 are_coplanar )
theorem Th23: :: PROJDES1:23
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( not b
2,b
3,b
4,b
5 are_coplanar & b
5,b
2,b
6 is_collinear & b
5,b
3,b
7 is_collinear & b
5,b
4,b
8 is_collinear & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 implies ( not b
6,b
7,b
8 is_collinear & not b
6,b
7,b
8,b
5 are_coplanar ) )
theorem Th24: :: PROJDES1:24
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13 being
Element of b
1 holds
not ( b
2,b
3,b
4,b
5 are_coplanar & not b
2,b
3,b
4,b
6 are_coplanar & not b
2,b
3,b
6,b
5 are_coplanar & not b
3,b
4,b
6,b
5 are_coplanar & not b
2,b
4,b
6,b
5 are_coplanar & b
5,b
6,b
7 is_collinear & b
5,b
2,b
8 is_collinear & b
5,b
3,b
9 is_collinear & b
5,b
4,b
10 is_collinear & b
2,b
6,b
11 is_collinear & b
8,b
7,b
11 is_collinear & b
3,b
6,b
12 is_collinear & b
9,b
7,b
12 is_collinear & b
4,b
6,b
13 is_collinear & b
5 <> b
8 & b
5 <> b
7 & b
6 <> b
7 & b
5 <> b
9 & b
11,b
12,b
13 is_collinear )
definition
let c
1 be
up-3-dimensional CollProjectiveSpace;
let c
2, c
3, c
4, c
5 be
Element of c
1;
pred c
2,c
3,c
4,c
5 constitute_a_quadrangle means :
Def2:
:: PROJDES1:def 2
( not a
3,a
4,a
5 is_collinear & not a
2,a
3,a
4 is_collinear & not a
2,a
4,a
5 is_collinear & not a
2,a
5,a
3 is_collinear );
end;
:: deftheorem Def2 defines constitute_a_quadrangle PROJDES1:def 2 :
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( b
2,b
3,b
4,b
5 constitute_a_quadrangle iff ( not b
3,b
4,b
5 is_collinear & not b
2,b
3,b
4 is_collinear & not b
2,b
4,b
5 is_collinear & not b
2,b
5,b
3 is_collinear ) );
Lemma24:
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of b1 holds
( b2 <> b3 & b2 <> b4 & b2 <> b5 & b6 <> b3 & b7 <> b4 & b2,b6,b7,b8 constitute_a_quadrangle & b2,b6,b3 is_collinear & b2,b7,b4 is_collinear & b2,b8,b5 is_collinear & b6,b7,b9 is_collinear & b3,b4,b9 is_collinear & b7,b8,b10 is_collinear & b4,b5,b10 is_collinear & b6,b8,b11 is_collinear & b3,b5,b11 is_collinear implies b9,b10,b11 is_collinear )
theorem Th25: :: PROJDES1:25
canceled;
theorem Th26: :: PROJDES1:26
for b
1 being
up-3-dimensional CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( not b
2,b
3,b
4 is_collinear & not b
2,b
4,b
5 is_collinear & not b
2,b
3,b
5 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 & b
3,b
4,b
9 is_collinear & b
6,b
7,b
9 is_collinear & b
3 <> b
6 & b
4,b
5,b
10 is_collinear & b
7,b
8,b
10 is_collinear & b
3,b
5,b
11 is_collinear & b
4 <> b
7 & b
6,b
8,b
11 is_collinear & b
2 <> b
6 & b
2 <> b
7 & b
2 <> b
8 implies b
10,b
11,b
9 is_collinear )
theorem Th27: :: PROJDES1:27