:: HESSENBE semantic presentation
theorem Th1: :: HESSENBE:1
canceled;
theorem Th2: :: HESSENBE:2
canceled;
theorem Th3: :: HESSENBE:3
for b
1 being
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 Th4: :: HESSENBE:4
theorem Th5: :: HESSENBE:5
for b
1 being
CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 & b
4,b
5,b
2 is_collinear & b
4,b
5,b
3 is_collinear & b
2,b
3,b
6 is_collinear implies b
4,b
5,b
6 is_collinear )
theorem Th6: :: HESSENBE:6
theorem Th7: :: HESSENBE:7
theorem Th8: :: HESSENBE:8
theorem Th9: :: HESSENBE:9
theorem Th10: :: HESSENBE:10
for b
1 being
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 )
theorem Th11: :: HESSENBE:11
canceled;
theorem Th12: :: HESSENBE:12
for b
1 being
CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( not b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
3,b
4,b
6 is_collinear & b
2,b
4,b
7 is_collinear & b
6,b
5,b
7 is_collinear & b
5 <> b
2 & b
5 <> b
3 & b
6 <> b
3 & b
6 <> b
4 implies ( b
2 <> b
7 & b
4 <> b
7 ) )
theorem Th13: :: HESSENBE:13
for b
1 being
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 is_collinear & b
2,b
3,b
5 is_collinear & b
4,b
6,b
5 is_collinear & b
6 <> b
4 & b
5 <> b
2 & b
6,b
2,b
4 is_collinear )
theorem Th14: :: HESSENBE:14
for b
1 being
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 is_collinear & b
2,b
3,b
5 is_collinear & b
4,b
5,b
6 is_collinear & b
2 <> b
5 & b
5 <> b
6 & b
3,b
2,b
6 is_collinear )
theorem Th15: :: HESSENBE:15
for b
1 being
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 is_collinear & b
2,b
3,b
5 is_collinear & b
4,b
6,b
5 is_collinear & b
5 <> b
6 & b
3 <> b
5 & b
5,b
3,b
6 is_collinear )
theorem Th16: :: HESSENBE:16
for b
1 being
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 is_collinear & b
2,b
3,b
5 is_collinear & b
4,b
6,b
2 is_collinear & b
2 <> b
5 & b
2 <> b
6 & b
5,b
2,b
6 is_collinear )
theorem Th17: :: HESSENBE:17
for b
1 being
CollProjectiveSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2 <> b
3 & b
4 <> b
5 & b
4,b
5,b
6 is_collinear & b
4,b
5,b
7 is_collinear & b
2,b
3,b
6 is_collinear & b
2,b
3,b
7 is_collinear & not b
2,b
3,b
4 is_collinear implies b
6 = b
7 )
theorem Th18: :: HESSENBE:18
canceled;
theorem Th19: :: HESSENBE:19
for b
1 being
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 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
4,b
6 is_collinear & b
2 <> b
5 & b
2 <> b
6 & b
2,b
5,b
6 is_collinear )
theorem Th20: :: HESSENBE:20
for b
1 being
Pappian CollProjectivePlanefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of b
1 holds
( b
2 <> b
3 & b
4 <> b
3 & b
5 <> b
6 & b
7 <> b
5 & b
7 <> b
6 & not b
4,b
2,b
7 is_collinear & b
4,b
2,b
3 is_collinear & b
7,b
5,b
6 is_collinear & b
4,b
5,b
8 is_collinear & b
7,b
2,b
8 is_collinear & b
4,b
6,b
9 is_collinear & b
3,b
7,b
9 is_collinear & b
2,b
6,b
10 is_collinear & b
3,b
5,b
10 is_collinear implies b
10,b
9,b
8 is_collinear )
Lemma14:
for b1 being Pappian CollProjectivePlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of b1 holds
( b2 <> b3 & b4 <> b3 & b2 <> b5 & b6 <> b5 & b2 <> b7 & b8 <> b7 & not b2,b4,b6 is_collinear & not b2,b4,b8 is_collinear & not b2,b6,b8 is_collinear & not b2,b9,b10 is_collinear & b4,b6,b10 is_collinear & b3,b5,b10 is_collinear & b6,b8,b9 is_collinear & b5,b7,b9 is_collinear & b4,b8,b11 is_collinear & b3,b7,b11 is_collinear & b2,b4,b3 is_collinear & b2,b6,b5 is_collinear & b2,b8,b7 is_collinear & ( b4,b6,b8 is_collinear or b3,b5,b7 is_collinear ) implies b9,b11,b10 is_collinear )
Lemma15:
for b1 being Pappian CollProjectivePlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of b1 holds
( b2 <> b3 & b4 <> b3 & b2 <> b5 & b6 <> b5 & b2 <> b7 & b8 <> b7 & not b2,b4,b6 is_collinear & not b2,b4,b8 is_collinear & not b2,b6,b8 is_collinear & not b2,b9,b10 is_collinear & b4,b6,b10 is_collinear & b3,b5,b10 is_collinear & b6,b8,b9 is_collinear & b5,b7,b9 is_collinear & b4,b8,b11 is_collinear & b3,b7,b11 is_collinear & b2,b4,b3 is_collinear & b2,b6,b5 is_collinear & b2,b8,b7 is_collinear & b2,b6,b12 is_collinear & b4,b8,b12 is_collinear & not b9,b10,b12 is_collinear implies b9,b11,b10 is_collinear )
Lemma16:
for b1 being Pappian CollProjectivePlane
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of b1 holds
( b2 <> b3 & b4 <> b3 & b2 <> b5 & b6 <> b5 & b2 <> b7 & b8 <> b7 & not b2,b4,b6 is_collinear & not b2,b4,b8 is_collinear & not b2,b6,b8 is_collinear & not b2,b9,b10 is_collinear & b4,b6,b10 is_collinear & b3,b5,b10 is_collinear & b6,b8,b9 is_collinear & b5,b7,b9 is_collinear & b4,b8,b11 is_collinear & b3,b7,b11 is_collinear & b2,b4,b3 is_collinear & b2,b6,b5 is_collinear & b2,b8,b7 is_collinear implies b9,b11,b10 is_collinear )
theorem Th21: :: HESSENBE:21
for b
1 being
Pappian CollProjectivePlanefor 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
3 & b
2 <> b
5 & b
6 <> b
5 & b
2 <> b
7 & b
8 <> b
7 & not b
2,b
4,b
6 is_collinear & not b
2,b
4,b
8 is_collinear & not b
2,b
6,b
8 is_collinear & b
4,b
6,b
9 is_collinear & b
3,b
5,b
9 is_collinear & b
6,b
8,b
10 is_collinear & b
5,b
7,b
10 is_collinear & b
4,b
8,b
11 is_collinear & b
3,b
7,b
11 is_collinear & b
2,b
4,b
3 is_collinear & b
2,b
6,b
5 is_collinear & b
2,b
8,b
7 is_collinear implies b
10,b
11,b
9 is_collinear )