:: PROJDES1 semantic presentation

theorem Th1: :: PROJDES1:1
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4 being Element of b1 holds
( b2,b3,b4 is_collinear implies ( b3,b4,b2 is_collinear & b4,b2,b3 is_collinear & b3,b2,b4 is_collinear & b2,b4,b3 is_collinear & b4,b3,b2 is_collinear ) )
proof end;

theorem Th2: :: PROJDES1:2
canceled;

theorem Th3: :: PROJDES1:3
canceled;

theorem Th4: :: PROJDES1:4
canceled;

theorem Th5: :: PROJDES1:5
for b1 being up-3-dimensional CollProjectiveSpace
for b2 being Element of b1 holds
not for b3, b4 being Element of b1 holds b2,b3,b4 is_collinear
proof end;

theorem Th6: :: PROJDES1:6
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
not ( not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2 <> b5 & b2,b5,b4 is_collinear )
proof end;

theorem Th7: :: PROJDES1:7
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
( not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b4,b5 is_collinear implies b2 = b5 )
proof end;

theorem Th8: :: PROJDES1:8
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
not ( not b2,b3,b4 is_collinear & b2,b4,b5 is_collinear & b3,b4,b6 is_collinear & b4 <> b5 & b7,b5,b6 is_collinear & b2,b3,b7 is_collinear & b2 <> b7 & not b6 <> b4 )
proof end;

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 )
proof end;

definition
let c1 be up-3-dimensional CollProjectiveSpace;
let c2, c3, c4, c5 be Element of c1;
pred c2,c3,c4,c5 are_coplanar means :Def1: :: PROJDES1:def 1
ex b1 being Element of a1 st
( a2,a3,b1 is_collinear & a4,a5,b1 is_collinear );
end;

:: deftheorem Def1 defines are_coplanar PROJDES1:def 1 :
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3,b4,b5 are_coplanar iff ex b6 being Element of b1 st
( b2,b3,b6 is_collinear & b4,b5,b6 is_collinear ) );

theorem Th9: :: PROJDES1:9
canceled;

theorem Th10: :: PROJDES1:10
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
( not ( not b2,b3,b4 is_collinear & not b3,b4,b5 is_collinear & not b4,b5,b2 is_collinear & not b5,b2,b3 is_collinear ) implies b2,b3,b4,b5 are_coplanar )
proof end;

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 )
proof end;

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 )
proof end;

theorem Th11: :: PROJDES1:11
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 & b4,b5,b2,b3 are_coplanar & b5,b2,b3,b4 are_coplanar & b3,b2,b4,b5 are_coplanar & b4,b3,b5,b2 are_coplanar & b5,b4,b2,b3 are_coplanar & b2,b5,b3,b4 are_coplanar & b2,b4,b5,b3 are_coplanar & b3,b5,b2,b4 are_coplanar & b4,b2,b3,b5 are_coplanar & b5,b3,b4,b2 are_coplanar & b4,b2,b5,b3 are_coplanar & b5,b3,b2,b4 are_coplanar & b2,b4,b3,b5 are_coplanar & b3,b5,b4,b2 are_coplanar & b2,b3,b5,b4 are_coplanar & b2,b5,b4,b3 are_coplanar & b3,b4,b2,b5 are_coplanar & b3,b2,b5,b4 are_coplanar & b4,b3,b2,b5 are_coplanar & b4,b5,b3,b2 are_coplanar & b5,b2,b4,b3 are_coplanar & b5,b4,b3,b2 are_coplanar ) )
proof end;

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 )
proof end;

theorem Th12: :: PROJDES1:12
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8 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 & b2,b3,b4,b8 are_coplanar implies b5,b6,b7,b8 are_coplanar )
proof end;

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 )
proof end;

theorem Th13: :: PROJDES1:13
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( not b2,b3,b4 is_collinear & b5,b6,b7,b2 are_coplanar & b5,b6,b7,b4 are_coplanar & b5,b6,b7,b3 are_coplanar & b2,b3,b4,b8 are_coplanar implies b5,b6,b7,b8 are_coplanar )
proof end;

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 )
proof end;

theorem Th14: :: PROJDES1:14
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds
( b2 <> b3 & b2,b3,b4 is_collinear & b5,b6,b7,b2 are_coplanar & b5,b6,b7,b3 are_coplanar implies b5,b6,b7,b4 are_coplanar )
proof end;

theorem Th15: :: PROJDES1:15
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
not ( not b2,b3,b4 is_collinear & b2,b3,b4,b5 are_coplanar & b2,b3,b4,b6 are_coplanar & b2,b3,b4,b7 are_coplanar & b2,b3,b4,b8 are_coplanar & ( for b9 being Element of b1 holds
not ( b5,b6,b9 is_collinear & b7,b8,b9 is_collinear ) ) )
proof end;

theorem Th16: :: PROJDES1:16
for b1 being up-3-dimensional CollProjectiveSpace holds
not for b2, b3, b4, b5 being Element of b1 holds b2,b3,b4,b5 are_coplanar
proof end;

theorem Th17: :: PROJDES1:17
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4 being Element of b1 holds
not ( not b2,b3,b4 is_collinear & ( for b5 being Element of b1 holds b2,b3,b4,b5 are_coplanar ) )
proof end;

theorem Th18: :: PROJDES1:18
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
( not ( not b2 = b3 & not b2 = b4 & not b3 = b4 & not b2 = b5 & not b3 = b5 & not b5 = b4 ) implies b2,b3,b4,b5 are_coplanar )
proof end;

theorem Th19: :: PROJDES1:19
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( not b2,b3,b4,b5 are_coplanar & b5,b2,b6 is_collinear & b2 <> b6 & b2,b3,b4,b6 are_coplanar )
proof end;

theorem Th20: :: PROJDES1:20
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of b1 holds
( not b2,b3,b4 is_collinear & not b5,b6,b7 is_collinear & b2,b3,b4,b8 are_coplanar & b2,b3,b4,b9 are_coplanar & b2,b3,b4,b10 are_coplanar & b5,b6,b7,b8 are_coplanar & b5,b6,b7,b9 are_coplanar & b5,b6,b7,b10 are_coplanar & not b2,b3,b4,b5 are_coplanar implies b8,b9,b10 is_collinear )
proof end;

theorem Th21: :: PROJDES1:21
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 & b4,b2,b3 is_collinear & not b2,b5,b6,b4 are_coplanar & not b3,b7,b8 is_collinear & b2,b5,b9 is_collinear & b3,b7,b9 is_collinear & b5,b6,b10 is_collinear & b7,b8,b10 is_collinear & b2,b6,b11 is_collinear & b3,b8,b11 is_collinear implies b9,b10,b11 is_collinear )
proof end;

theorem Th22: :: PROJDES1:22
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6 being Element of b1 holds
not ( not b2,b3,b4,b5 are_coplanar & b2,b3,b4,b6 are_coplanar & not b2,b3,b6 is_collinear & b2,b3,b5,b6 are_coplanar )
proof end;

theorem Th23: :: PROJDES1:23
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8 being Element of b1 holds
( not b2,b3,b4,b5 are_coplanar & b5,b2,b6 is_collinear & b5,b3,b7 is_collinear & b5,b4,b8 is_collinear & b5 <> b6 & b5 <> b7 & b5 <> b8 implies ( not b6,b7,b8 is_collinear & not b6,b7,b8,b5 are_coplanar ) )
proof end;

theorem Th24: :: PROJDES1:24
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13 being Element of b1 holds
not ( b2,b3,b4,b5 are_coplanar & not b2,b3,b4,b6 are_coplanar & not b2,b3,b6,b5 are_coplanar & not b3,b4,b6,b5 are_coplanar & not b2,b4,b6,b5 are_coplanar & b5,b6,b7 is_collinear & b5,b2,b8 is_collinear & b5,b3,b9 is_collinear & b5,b4,b10 is_collinear & b2,b6,b11 is_collinear & b8,b7,b11 is_collinear & b3,b6,b12 is_collinear & b9,b7,b12 is_collinear & b4,b6,b13 is_collinear & b5 <> b8 & b5 <> b7 & b6 <> b7 & b5 <> b9 & b11,b12,b13 is_collinear )
proof end;

definition
let c1 be up-3-dimensional CollProjectiveSpace;
let c2, c3, c4, c5 be Element of c1;
pred c2,c3,c4,c5 constitute_a_quadrangle means :Def2: :: PROJDES1:def 2
( not a3,a4,a5 is_collinear & not a2,a3,a4 is_collinear & not a2,a4,a5 is_collinear & not a2,a5,a3 is_collinear );
end;

:: deftheorem Def2 defines constitute_a_quadrangle PROJDES1:def 2 :
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5 being Element of b1 holds
( b2,b3,b4,b5 constitute_a_quadrangle iff ( not b3,b4,b5 is_collinear & not b2,b3,b4 is_collinear & not b2,b4,b5 is_collinear & not b2,b5,b3 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 )
proof end;

theorem Th25: :: PROJDES1:25
canceled;

theorem Th26: :: PROJDES1:26
for b1 being up-3-dimensional CollProjectiveSpace
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of b1 holds
( not b2,b3,b4 is_collinear & not b2,b4,b5 is_collinear & not b2,b3,b5 is_collinear & b2,b3,b6 is_collinear & b2,b4,b7 is_collinear & b2,b5,b8 is_collinear & b3,b4,b9 is_collinear & b6,b7,b9 is_collinear & b3 <> b6 & b4,b5,b10 is_collinear & b7,b8,b10 is_collinear & b3,b5,b11 is_collinear & b4 <> b7 & b6,b8,b11 is_collinear & b2 <> b6 & b2 <> b7 & b2 <> b8 implies b10,b11,b9 is_collinear )
proof end;

theorem Th27: :: PROJDES1:27
for b1 being up-3-dimensional CollProjectiveSpace holds b1 is Desarguesian
proof end;

registration
cluster up-3-dimensional -> Desarguesian up-3-dimensional L5();
correctness
coherence
for b1 being up-3-dimensional CollProjectiveSpace holds b1 is Desarguesian
;
by Th27;
end;