:: ANPROJ_2 semantic presentation
theorem Th1: :: ANPROJ_2:1
Lemma2:
for b1 being RealLinearSpace
for b2, b3 being Element of b1 holds
( ( for b4, b5 being Real holds
( (b4 * b2) + (b5 * b3) = 0. b1 implies ( b4 = 0 & b5 = 0 ) ) ) implies ( b2 is_Prop_Vect & b3 is_Prop_Vect & not are_Prop b2,b3 ) )
theorem Th2: :: ANPROJ_2:2
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( ( for b
6, b
7, b
8, b
9 being
Real holds
(
(((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b
1 implies ( b
6 = 0 & b
7 = 0 & b
8 = 0 & b
9 = 0 ) ) ) implies ( b
2 is_Prop_Vect & b
3 is_Prop_Vect & not
are_Prop b
2,b
3 & b
4 is_Prop_Vect & b
5 is_Prop_Vect & not
are_Prop b
4,b
5 & not b
2,b
3,b
4 are_LinDep & not b
4,b
5,b
2 are_LinDep ) )
Lemma4:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1
for b5, b6, b7, b8 being Real holds b5 * (((b6 * b2) + (b7 * b3)) + (b8 * b4)) = (((b5 * b6) * b2) + ((b5 * b7) * b3)) + ((b5 * b8) * b4)
Lemma5:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds ((b2 + b3) + b4) + ((b5 + b6) + b7) = ((b2 + b5) + (b3 + b6)) + (b4 + b7)
theorem Th3: :: ANPROJ_2:3
for b
1 being
RealLinearSpacefor b
2, b
3, b
4 being
Element of b
1 holds
( ( for b
5 being
Element of b
1 holds
ex b
6, b
7, b
8 being
Real st b
5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4) ) & ( for b
5, b
6, b
7 being
Real holds
(
((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b
1 implies ( b
5 = 0 & b
6 = 0 & b
7 = 0 ) ) ) implies for b
5, b
6 being
Element of b
1 holds
ex b
7 being
Element of b
1 st
( b
2,b
3,b
7 are_LinDep & b
5,b
6,b
7 are_LinDep & b
7 is_Prop_Vect ) )
Lemma7:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being Element of b1
for b6, b7, b8, b9, b10 being Real holds b6 * ((((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) = ((((b6 * b7) * b2) + ((b6 * b8) * b3)) + ((b6 * b9) * b4)) + ((b6 * b10) * b5)
Lemma8:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of b1 holds (((b2 + b3) + b4) + b5) + (((b6 + b7) + b8) + b9) = (((b2 + b6) + (b3 + b7)) + (b4 + b8)) + (b5 + b9)
Lemma9:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1
for b5, b6, b7, b8 being Real holds b5 * (((b6 * b2) + (b7 * b3)) + (b8 * b4)) = (((b5 * b6) * b2) + ((b5 * b7) * b3)) + ((b5 * b8) * b4)
Lemma10:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6 being Element of b1
for b7, b8, b9, b10, b11 being Real holds
( b2 = (b7 * b3) + (b8 * b4) & b4 = ((b9 * b3) + (b10 * b5)) + (b11 * b6) implies b2 = (((b7 + (b8 * b9)) * b3) + ((b8 * b10) * b5)) + ((b8 * b11) * b6) )
theorem Th4: :: ANPROJ_2:4
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( ( for b
6 being
Element of b
1 holds
ex b
7, b
8, b
9, b
10 being
Real st b
6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5) ) & ( for b
6, b
7, b
8, b
9 being
Real holds
(
(((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b
1 implies ( b
6 = 0 & b
7 = 0 & b
8 = 0 & b
9 = 0 ) ) ) implies for b
6, b
7 being
Element of b
1 holds
not ( b
6 is_Prop_Vect & b
7 is_Prop_Vect & ( for b
8, b
9 being
Element of b
1 holds
not ( b
6,b
7,b
9 are_LinDep & b
3,b
4,b
8 are_LinDep & b
2,b
9,b
8 are_LinDep & b
8 is_Prop_Vect & b
9 is_Prop_Vect ) ) ) )
theorem Th5: :: ANPROJ_2:5
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
Element of b
1 holds
( ( for b
6, b
7, b
8, b
9 being
Real holds
(
(((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b
1 implies ( b
6 = 0 & b
7 = 0 & b
8 = 0 & b
9 = 0 ) ) ) implies for b
6 being
Element of b
1 holds
not ( b
6 is_Prop_Vect & b
2,b
3,b
6 are_LinDep & b
4,b
5,b
6 are_LinDep ) )
:: deftheorem Def1 defines are_Prop_Vect ANPROJ_2:def 1 :
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4, c
5, c
6, c
7 be
Element of c
1;
pred c
2,c
3,c
4,c
5,c
6,c
7 lie_on_a_triangle means :
Def2:
:: ANPROJ_2:def 2
( a
2,a
3,a
7 are_LinDep & a
2,a
4,a
6 are_LinDep & a
3,a
4,a
5 are_LinDep );
end;
:: deftheorem Def2 defines lie_on_a_triangle ANPROJ_2:def 2 :
for b
1 being
RealLinearSpacefor 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
6,b
7 lie_on_a_triangle iff ( b
2,b
3,b
7 are_LinDep & b
2,b
4,b
6 are_LinDep & b
3,b
4,b
5 are_LinDep ) );
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
Element of c
1;
pred c
2,c
3,c
4,c
5,c
6,c
7,c
8 are_perspective means :
Def3:
:: ANPROJ_2:def 3
( a
2,a
3,a
6 are_LinDep & a
2,a
4,a
7 are_LinDep & a
2,a
5,a
8 are_LinDep );
end;
:: deftheorem Def3 defines are_perspective ANPROJ_2:def 3 :
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3,b
4,b
5,b
6,b
7,b
8 are_perspective iff ( b
2,b
3,b
6 are_LinDep & b
2,b
4,b
7 are_LinDep & b
2,b
5,b
8 are_LinDep ) );
Lemma16:
for b1 being RealLinearSpace
for b2 being Element of b1
for b3 being Real holds - (b3 * b2) = (- b3) * b2
theorem Th6: :: ANPROJ_2:6
for b
1 being
RealLinearSpacefor b
2, b
3, b
4 being
Element of b
1 holds
( b
2,b
3,b
4 are_LinDep & not
are_Prop b
2,b
3 & not
are_Prop b
2,b
4 & not
are_Prop b
3,b
4 & b
2,b
3,b
4 are_Prop_Vect implies ( ex b
5, b
6 being
Real st
( b
6 * b
4 = b
2 + (b5 * b3) & b
5 <> 0 & b
6 <> 0 ) & ex b
5, b
6 being
Real st
( b
4 = (b6 * b2) + (b5 * b3) & b
6 <> 0 & b
5 <> 0 ) ) )
theorem Th7: :: ANPROJ_2:7
Lemma19:
for b1 being RealLinearSpace
for b2, b3 being Element of b1
for b4 being Real holds
( b4 * b2 = b3 & b4 <> 0 implies are_Prop b2,b3 )
Lemma20:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6 being Element of b1
for b7, b8 being Real holds
( b2 = b3 + (b7 * b4) & b5 = b3 + (b8 * b6) & are_Prop b2,b5 & b7 <> 0 implies b3,b4,b6 are_LinDep )
Lemma21:
for b1 being RealLinearSpace
for b2, b3 being Element of b1
for b4 being Real holds
( b4 * b2 = b3 & b4 <> 0 & b2 is_Prop_Vect implies b3 is_Prop_Vect )
Lemma22:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7 being Element of b1
for b8, b9, b10, b11 being Real holds
( b2 = (b10 * b3) + (b11 * b4) & b3 = b5 + (b8 * b6) & b4 = b5 + (b9 * b7) implies b2 = (((b10 + b11) * b5) + ((b10 * b8) * b6)) + ((b11 * b9) * b7) )
Lemma23:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being Element of b1
for b6, b7 being Real holds
( b2 = (b6 * b3) + (b7 * b4) implies b2 = ((0 * b5) + (b6 * b3)) + (b7 * b4) )
Lemma24:
for b1 being RealLinearSpace
for b2, b3 being Element of b1 holds (0 * b2) + (0 * b3) = 0. b1
Lemma25:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1
for b5, b6, b7 being Real holds ((0 * b2) + ((b5 * b6) * b3)) + (((- b5) * b7) * b4) = b5 * ((b6 * b3) - (b7 * b4))
theorem Th8: :: ANPROJ_2:8
for b
1 being
RealLinearSpacefor 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 is_Prop_Vect & b
3,b
4,b
5 are_Prop_Vect & b
6,b
7,b
8 are_Prop_Vect & b
9,b
10,b
11 are_Prop_Vect & b
2,b
3,b
4,b
5,b
6,b
7,b
8 are_perspective & not
are_Prop b
2,b
6 & not
are_Prop b
2,b
7 & not
are_Prop b
2,b
8 & not
are_Prop b
3,b
6 & not
are_Prop b
4,b
7 & not
are_Prop b
5,b
8 & not b
2,b
3,b
4 are_LinDep & not b
2,b
3,b
5 are_LinDep & not b
2,b
4,b
5 are_LinDep & b
3,b
4,b
5,b
9,b
10,b
11 lie_on_a_triangle & b
6,b
7,b
8,b
9,b
10,b
11 lie_on_a_triangle implies b
9,b
10,b
11 are_LinDep )
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
Element of c
1;
pred c
2,c
3,c
4,c
5,c
6,c
7,c
8 lie_on_an_angle means :
Def4:
:: ANPROJ_2:def 4
( not a
2,a
3,a
6 are_LinDep & a
2,a
3,a
4 are_LinDep & a
2,a
3,a
5 are_LinDep & a
2,a
6,a
7 are_LinDep & a
2,a
6,a
8 are_LinDep );
end;
:: deftheorem Def4 defines lie_on_an_angle ANPROJ_2:def 4 :
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3,b
4,b
5,b
6,b
7,b
8 lie_on_an_angle iff ( not b
2,b
3,b
6 are_LinDep & b
2,b
3,b
4 are_LinDep & b
2,b
3,b
5 are_LinDep & b
2,b
6,b
7 are_LinDep & b
2,b
6,b
8 are_LinDep ) );
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
Element of c
1;
pred c
2,c
3,c
4,c
5,c
6,c
7,c
8 are_half_mutually_not_Prop means :
Def5:
:: ANPROJ_2:def 5
( not
are_Prop a
2,a
4 & not
are_Prop a
2,a
5 & not
are_Prop a
2,a
7 & not
are_Prop a
2,a
8 & not
are_Prop a
3,a
4 & not
are_Prop a
3,a
5 & not
are_Prop a
6,a
7 & not
are_Prop a
6,a
8 & not
are_Prop a
4,a
5 & not
are_Prop a
7,a
8 );
end;
:: deftheorem Def5 defines are_half_mutually_not_Prop ANPROJ_2:def 5 :
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of b
1 holds
( b
2,b
3,b
4,b
5,b
6,b
7,b
8 are_half_mutually_not_Prop iff ( not
are_Prop b
2,b
4 & not
are_Prop b
2,b
5 & not
are_Prop b
2,b
7 & not
are_Prop b
2,b
8 & not
are_Prop b
3,b
4 & not
are_Prop b
3,b
5 & not
are_Prop b
6,b
7 & not
are_Prop b
6,b
8 & not
are_Prop b
4,b
5 & not
are_Prop b
7,b
8 ) );
Lemma29:
for b1 being RealLinearSpace
for b2, b3 being Element of b1
for b4 being Real holds
( b4 * b2 = b3 & b4 <> 0 implies are_Prop b2,b3 )
Lemma30:
for b1 being RealLinearSpace
for b2, b3, b4 being Element of b1
for b5 being Real holds
not ( not are_Prop b2,b3 & b4 = b5 * b3 & b5 <> 0 & are_Prop b2,b4 )
Lemma31:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6 being Element of b1
for b7, b8, b9 being Real holds
( b2 = (b7 * b3) + (b8 * b4) & b4 = b5 + (b9 * b6) implies b2 = ((b8 * b5) + (b7 * b3)) + ((b8 * b9) * b6) )
Lemma32:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6 being Element of b1
for b7, b8, b9 being Real holds
( b2 = (b7 * b3) + (b8 * b4) & b4 = b5 + (b9 * b6) implies b2 = ((b8 * b5) + ((b8 * b9) * b6)) + (b7 * b3) )
Lemma33:
for b1 being RealLinearSpace
for b2, b3 being Element of b1
for b4 being Real holds
( b4 * b2 = b3 & b4 <> 0 & b2 is_Prop_Vect implies b3 is_Prop_Vect )
Lemma34:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being Element of b1
for b6, b7 being Real holds
not ( not are_Prop b2,b3 & b4 = b6 * b3 & b6 <> 0 & b5 = b7 * b2 & b7 <> 0 & are_Prop b5,b4 )
Lemma35:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7 being Element of b1
for b8, b9, b10, b11 being Real holds
( b2 = (b10 * b3) + (b11 * b4) & b3 = b5 + (b8 * b6) & b4 = b5 + (b9 * b7) implies b2 = (((b10 + b11) * b5) + ((b10 * b8) * b6)) + ((b11 * b9) * b7) )
Lemma36:
for b1, b2, b3 being Real holds
not ( b1 <> b2 & b3 <> 0 & not (b2 * b3) - (b1 * b3) <> 0 )
Lemma37:
for b1, b2, b3, b4, b5, b6, b7, b8 being Real holds
( b5 + b7 = b6 + b8 & b5 * b1 = b6 * b2 & b7 * b3 = b8 * b4 & b1 <> b2 & b4 <> 0 implies b5 = (((b2 * b3) - (b2 * b4)) * (((b2 * b4) - (b1 * b4)) " )) * b7 )
Lemma38:
for b1, b2, b3, b4 being Real holds
not ( b1 <> 0 & b2 <> b3 & b4 <> 0 & not b4 * (((b3 * b1) - (b2 * b1)) " ) <> 0 )
Lemma39:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being Element of b1
for b6, b7, b8, b9, b10, b11 being Real holds
( b10 = (((b6 * b7) - (b6 * b8)) * (((b6 * b8) - (b9 * b8)) " )) * b11 & b8 <> 0 & b9 <> b6 & b2 = (((b10 + b11) * b3) + ((b10 * b9) * b4)) + ((b11 * b7) * b5) implies b2 = (b11 * (((b6 * b8) - (b9 * b8)) " )) * (((((b6 * b7) - (b9 * b8)) * b3) + (((b9 * b6) * (b7 - b8)) * b4)) + (((b8 * b7) * (b6 - b9)) * b5)) )
Lemma40:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7 being Element of b1 holds ((b2 + b3) + b4) + ((b5 + b6) + b7) = ((b2 + b5) + (b3 + b6)) + (b4 + b7)
Lemma41:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7 being Element of b1
for b8, b9, b10, b11, b12, b13 being Real holds
( b2 = ((((b8 * b9) - (b10 * b11)) * b3) + (((b10 * b8) * (b9 - b11)) * b4)) + (((b11 * b9) * (b8 - b10)) * b5) & b6 = (b3 + (b8 * b4)) + (b9 * b5) & b7 = (b3 + (b10 * b4)) + (b11 * b5) & b12 + b13 = (b10 * b11) - (b8 * b9) & (b12 * b8) + (b13 * b10) = (b10 * b8) * (b11 - b9) & (b12 * b9) + (b13 * b11) = (b11 * b9) * (b10 - b8) implies ((1 * b2) + (b12 * b6)) + (b13 * b7) = 0. b1 )
Lemma42:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6 being Element of b1
for b7, b8, b9, b10, b11, b12 being Real holds
( b2 = (b3 + (b7 * b4)) + (b8 * b5) & b6 = ((b11 * b3) + (b9 * b4)) + ((b11 * b8) * b5) & b11 = b12 & b9 = b12 * b7 & b11 * b8 = b10 implies b6 = b11 * b2 )
theorem Th9: :: ANPROJ_2:9
for b
1 being
RealLinearSpacefor 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 is_Prop_Vect & b
3,b
4,b
5 are_Prop_Vect & b
6,b
7,b
8 are_Prop_Vect & b
9,b
10,b
11 are_Prop_Vect & b
2,b
3,b
4,b
5,b
6,b
7,b
8 lie_on_an_angle & b
2,b
3,b
4,b
5,b
6,b
7,b
8 are_half_mutually_not_Prop & b
3,b
7,b
11 are_LinDep & b
6,b
4,b
11 are_LinDep & b
3,b
8,b
10 are_LinDep & b
5,b
6,b
10 are_LinDep & b
4,b
8,b
9 are_LinDep & b
5,b
7,b
9 are_LinDep implies b
9,b
10,b
11 are_LinDep )
theorem Th10: :: ANPROJ_2:10
for b
1 being non
empty set for b
2, b
3, b
4 being
Element of b
1 holds
ex b
5, b
6, b
7 being
Element of
Funcs b
1,
REAL st
( ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
2 implies b
5 . b
8 = 1 ) & ( b
8 <> b
2 implies b
5 . b
8 = 0 ) ) ) ) & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
3 implies b
6 . b
8 = 1 ) & ( b
8 <> b
3 implies b
6 . b
8 = 0 ) ) ) ) & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
4 implies b
7 . b
8 = 1 ) & ( b
8 <> b
4 implies b
7 . b
8 = 0 ) ) ) ) )
theorem Th11: :: ANPROJ_2:11
for b
1 being non
empty set for b
2, b
3, b
4 being
Element of
Funcs b
1,
REAL for b
5, b
6, b
7 being
Element of b
1 holds
( b
5 in b
1 & b
6 in b
1 & b
7 in b
1 & b
5 <> b
6 & b
5 <> b
7 & b
6 <> b
7 & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
5 implies b
2 . b
8 = 1 ) & ( b
8 <> b
5 implies b
2 . b
8 = 0 ) ) ) ) & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
6 implies b
3 . b
8 = 1 ) & ( b
8 <> b
6 implies b
3 . b
8 = 0 ) ) ) ) & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
7 implies b
4 . b
8 = 1 ) & ( b
8 <> b
7 implies b
4 . b
8 = 0 ) ) ) ) implies for b
8, b
9, b
10 being
Real holds
(
(RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b8,b2]),((RealFuncExtMult b1) . [b9,b3])),
((RealFuncExtMult b1) . [b10,b4]) = RealFuncZero b
1 implies ( b
8 = 0 & b
9 = 0 & b
10 = 0 ) ) )
theorem Th12: :: ANPROJ_2:12
for b
1 being non
empty set for b
2, b
3, b
4 being
Element of b
1 holds
not ( b
2 in b
1 & b
3 in b
1 & b
4 in b
1 & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 & ( for b
5, b
6, b
7 being
Element of
Funcs b
1,
REAL holds
ex b
8, b
9, b
10 being
Real st
(
(RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b8,b5]),((RealFuncExtMult b1) . [b9,b6])),
((RealFuncExtMult b1) . [b10,b7]) = RealFuncZero b
1 & not ( b
8 = 0 & b
9 = 0 & b
10 = 0 ) ) ) )
theorem Th13: :: ANPROJ_2:13
for b
1 being non
empty set for b
2, b
3, b
4 being
Element of
Funcs b
1,
REAL for b
5, b
6, b
7 being
Element of b
1 holds
( b
1 = {b5,b6,b7} & b
5 <> b
6 & b
5 <> b
7 & b
6 <> b
7 & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
5 implies b
2 . b
8 = 1 ) & ( b
8 <> b
5 implies b
2 . b
8 = 0 ) ) ) ) & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
6 implies b
3 . b
8 = 1 ) & ( b
8 <> b
6 implies b
3 . b
8 = 0 ) ) ) ) & ( for b
8 being
set holds
( b
8 in b
1 implies ( ( b
8 = b
7 implies b
4 . b
8 = 1 ) & ( b
8 <> b
7 implies b
4 . b
8 = 0 ) ) ) ) implies for b
8 being
Element of
Funcs b
1,
REAL holds
ex b
9, b
10, b
11 being
Real st b
8 = (RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b9,b2]),((RealFuncExtMult b1) . [b10,b3])),
((RealFuncExtMult b1) . [b11,b4]) )
theorem Th14: :: ANPROJ_2:14
for b
1 being non
empty set for b
2, b
3, b
4 being
Element of b
1 holds
not ( b
1 = {b2,b3,b4} & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 & ( for b
5, b
6, b
7 being
Element of
Funcs b
1,
REAL holds
ex b
8 being
Element of
Funcs b
1,
REAL st
for b
9, b
10, b
11 being
Real holds
not b
8 = (RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b9,b5]),((RealFuncExtMult b1) . [b10,b6])),
((RealFuncExtMult b1) . [b11,b7]) ) )
theorem Th15: :: ANPROJ_2:15
for b
1 being non
empty set for b
2, b
3, b
4 being
Element of b
1 holds
not ( b
1 = {b2,b3,b4} & b
2 <> b
3 & b
2 <> b
4 & b
3 <> b
4 & ( for b
5, b
6, b
7 being
Element of
Funcs b
1,
REAL holds
not ( ( for b
8, b
9, b
10 being
Real holds
(
(RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b8,b5]),((RealFuncExtMult b1) . [b9,b6])),
((RealFuncExtMult b1) . [b10,b7]) = RealFuncZero b
1 implies ( b
8 = 0 & b
9 = 0 & b
10 = 0 ) ) ) & ( for b
8 being
Element of
Funcs b
1,
REAL holds
ex b
9, b
10, b
11 being
Real st b
8 = (RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b9,b5]),((RealFuncExtMult b1) . [b10,b6])),
((RealFuncExtMult b1) . [b11,b7]) ) ) ) )
Lemma48:
ex b1 being non empty set ex b2, b3, b4 being Element of b1 st
( b1 = {b2,b3,b4} & b2 <> b3 & b2 <> b4 & b3 <> b4 )
theorem Th16: :: ANPROJ_2:16
ex b
1 being non
trivial RealLinearSpaceex b
2, b
3, b
4 being
Element of b
1 st
( ( for b
5, b
6, b
7 being
Real holds
(
((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b
1 implies ( b
5 = 0 & b
6 = 0 & b
7 = 0 ) ) ) & ( for b
5 being
Element of b
1 holds
ex b
6, b
7, b
8 being
Real st b
5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4) ) )
theorem Th17: :: ANPROJ_2:17
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of b
1 holds
ex b
6, b
7, b
8, b
9 being
Element of
Funcs b
1,
REAL st
( ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
2 implies b
6 . b
10 = 1 ) & ( b
10 <> b
2 implies b
6 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
3 implies b
7 . b
10 = 1 ) & ( b
10 <> b
3 implies b
7 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
4 implies b
8 . b
10 = 1 ) & ( b
10 <> b
4 implies b
8 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
5 implies b
9 . b
10 = 1 ) & ( b
10 <> b
5 implies b
9 . b
10 = 0 ) ) ) ) )
theorem Th18: :: ANPROJ_2:18
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of
Funcs b
1,
REAL for b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
6 in b
1 & b
7 in b
1 & b
8 in b
1 & b
9 in b
1 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
6 implies b
2 . b
10 = 1 ) & ( b
10 <> b
6 implies b
2 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
7 implies b
3 . b
10 = 1 ) & ( b
10 <> b
7 implies b
3 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
8 implies b
4 . b
10 = 1 ) & ( b
10 <> b
8 implies b
4 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
9 implies b
5 . b
10 = 1 ) & ( b
10 <> b
9 implies b
5 . b
10 = 0 ) ) ) ) implies for b
10, b
11, b
12, b
13 being
Real holds
(
(RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b10,b2]),((RealFuncExtMult b1) . [b11,b3])),((RealFuncExtMult b1) . [b12,b4])),
((RealFuncExtMult b1) . [b13,b5]) = RealFuncZero b
1 implies ( b
10 = 0 & b
11 = 0 & b
12 = 0 & b
13 = 0 ) ) )
theorem Th19: :: ANPROJ_2:19
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
2 in b
1 & b
3 in b
1 & b
4 in b
1 & b
5 in b
1 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
3 <> b
4 & b
3 <> b
5 & b
4 <> b
5 & ( for b
6, b
7, b
8, b
9 being
Element of
Funcs b
1,
REAL holds
ex b
10, b
11, b
12, b
13 being
Real st
(
(RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b10,b6]),((RealFuncExtMult b1) . [b11,b7])),((RealFuncExtMult b1) . [b12,b8])),
((RealFuncExtMult b1) . [b13,b9]) = RealFuncZero b
1 & not ( b
10 = 0 & b
11 = 0 & b
12 = 0 & b
13 = 0 ) ) ) )
theorem Th20: :: ANPROJ_2:20
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of
Funcs b
1,
REAL for b
6, b
7, b
8, b
9 being
Element of b
1 holds
( b
1 = {b6,b7,b8,b9} & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
6 implies b
2 . b
10 = 1 ) & ( b
10 <> b
6 implies b
2 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
7 implies b
3 . b
10 = 1 ) & ( b
10 <> b
7 implies b
3 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
8 implies b
4 . b
10 = 1 ) & ( b
10 <> b
8 implies b
4 . b
10 = 0 ) ) ) ) & ( for b
10 being
set holds
( b
10 in b
1 implies ( ( b
10 = b
9 implies b
5 . b
10 = 1 ) & ( b
10 <> b
9 implies b
5 . b
10 = 0 ) ) ) ) implies for b
10 being
Element of
Funcs b
1,
REAL holds
ex b
11, b
12, b
13, b
14 being
Real st b
10 = (RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b11,b2]),((RealFuncExtMult b1) . [b12,b3])),((RealFuncExtMult b1) . [b13,b4])),
((RealFuncExtMult b1) . [b14,b5]) )
theorem Th21: :: ANPROJ_2:21
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
1 = {b2,b3,b4,b5} & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
3 <> b
4 & b
3 <> b
5 & b
4 <> b
5 & ( for b
6, b
7, b
8, b
9 being
Element of
Funcs b
1,
REAL holds
ex b
10 being
Element of
Funcs b
1,
REAL st
for b
11, b
12, b
13, b
14 being
Real holds
not b
10 = (RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b11,b6]),((RealFuncExtMult b1) . [b12,b7])),((RealFuncExtMult b1) . [b13,b8])),
((RealFuncExtMult b1) . [b14,b9]) ) )
theorem Th22: :: ANPROJ_2:22
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of b
1 holds
not ( b
1 = {b2,b3,b4,b5} & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
3 <> b
4 & b
3 <> b
5 & b
4 <> b
5 & ( for b
6, b
7, b
8, b
9 being
Element of
Funcs b
1,
REAL holds
not ( ( for b
10, b
11, b
12, b
13 being
Real holds
(
(RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b10,b6]),((RealFuncExtMult b1) . [b11,b7])),((RealFuncExtMult b1) . [b12,b8])),
((RealFuncExtMult b1) . [b13,b9]) = RealFuncZero b
1 implies ( b
10 = 0 & b
11 = 0 & b
12 = 0 & b
13 = 0 ) ) ) & ( for b
10 being
Element of
Funcs b
1,
REAL holds
ex b
11, b
12, b
13, b
14 being
Real st b
10 = (RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncAdd b1) . ((RealFuncExtMult b1) . [b11,b6]),((RealFuncExtMult b1) . [b12,b7])),((RealFuncExtMult b1) . [b13,b8])),
((RealFuncExtMult b1) . [b14,b9]) ) ) ) )
Lemma54:
ex b1 being non empty set ex b2, b3, b4, b5 being Element of b1 st
( b1 = {b2,b3,b4,b5} & b2 <> b3 & b2 <> b4 & b2 <> b5 & b3 <> b4 & b3 <> b5 & b4 <> b5 )
theorem Th23: :: ANPROJ_2:23
ex b
1 being non
trivial RealLinearSpaceex b
2, b
3, b
4, b
5 being
Element of b
1 st
( ( for b
6, b
7, b
8, b
9 being
Real holds
(
(((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b
1 implies ( b
6 = 0 & b
7 = 0 & b
8 = 0 & b
9 = 0 ) ) ) & ( for b
6 being
Element of b
1 holds
ex b
7, b
8, b
9, b
10 being
Real st b
6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5) ) )
:: deftheorem Def6 defines up-3-dimensional ANPROJ_2:def 6 :
definition
let c
1 be non
empty CollStr ;
redefine attr a
1 is
reflexive means :
Def7:
:: ANPROJ_2:def 7
for b
1, b
2, b
3 being
Element of a
1 holds
( b
1,b
2,b
1 is_collinear & b
1,b
1,b
2 is_collinear & b
1,b
2,b
2 is_collinear );
compatibility
( c1 is reflexive iff for b1, b2, b3 being Element of c1 holds
( b1,b2,b1 is_collinear & b1,b1,b2 is_collinear & b1,b2,b2 is_collinear ) )
redefine attr a
1 is
transitive means :
Def8:
:: ANPROJ_2:def 8
for b
1, b
2, b
3, b
4, b
5 being
Element of a
1 holds
( b
1 <> b
2 & b
1,b
2,b
3 is_collinear & b
1,b
2,b
4 is_collinear & b
1,b
2,b
5 is_collinear implies b
3,b
4,b
5 is_collinear );
compatibility
( c1 is transitive iff for b1, b2, b3, b4, b5 being Element of c1 holds
( b1 <> b2 & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b2,b5 is_collinear implies b3,b4,b5 is_collinear ) )
end;
:: deftheorem Def7 defines reflexive ANPROJ_2:def 7 :
:: deftheorem Def8 defines transitive ANPROJ_2:def 8 :
for b
1 being non
empty CollStr holds
( b
1 is
transitive iff for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 <> b
3 & b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
3,b
6 is_collinear implies b
4,b
5,b
6 is_collinear ) );
definition
let c
1 be non
empty CollStr ;
attr a
1 is
Vebleian means :
Def9:
:: ANPROJ_2:def 9
for b
1, b
2, b
3, b
4, b
5 being
Element of a
1 holds
not ( b
1,b
2,b
4 is_collinear & b
2,b
3,b
5 is_collinear & ( for b
6 being
Element of a
1 holds
not ( b
1,b
3,b
6 is_collinear & b
4,b
5,b
6 is_collinear ) ) );
attr a
1 is
at_least_3rank means :
Def10:
:: ANPROJ_2:def 10
for b
1, b
2 being
Element of a
1 holds
ex b
3 being
Element of a
1 st
( b
1 <> b
3 & b
2 <> b
3 & b
1,b
2,b
3 is_collinear );
end;
:: deftheorem Def9 defines Vebleian ANPROJ_2:def 9 :
for b
1 being non
empty CollStr holds
( b
1 is
Vebleian iff for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( b
2,b
3,b
5 is_collinear & b
3,b
4,b
6 is_collinear & ( for b
7 being
Element of b
1 holds
not ( b
2,b
4,b
7 is_collinear & b
5,b
6,b
7 is_collinear ) ) ) );
:: deftheorem Def10 defines at_least_3rank ANPROJ_2:def 10 :
theorem Th24: :: ANPROJ_2:24
Lemma62:
for b1 being non trivial RealLinearSpace holds ProjectiveSpace b1 is reflexive
Lemma63:
for b1 being non trivial RealLinearSpace holds ProjectiveSpace b1 is transitive
theorem Th25: :: ANPROJ_2:25
for b
1 being non
trivial RealLinearSpacefor b
2, b
3, b
4 being
Element of
(ProjectiveSpace b1) 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
4,b
3,b
2 is_collinear & b
4,b
2,b
3 is_collinear & b
3,b
4,b
2 is_collinear ) )
Lemma65:
for b1 being non trivial RealLinearSpace
for b2, b3, b4, b5, b6 being Element of (ProjectiveSpace b1) holds
not ( b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b3,b4,b6 is_collinear & ( for b7 being Element of (ProjectiveSpace b1) holds
not ( b2,b4,b7 is_collinear & b5,b6,b7 is_collinear ) ) )
Lemma66:
for b1 being non trivial RealLinearSpace
for b2, b3, b4, b5, b6 being Element of (ProjectiveSpace b1) holds
not ( not b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b3,b4,b6 is_collinear & ( for b7 being Element of (ProjectiveSpace b1) holds
not ( b2,b4,b7 is_collinear & b5,b6,b7 is_collinear ) ) )
Lemma67:
for b1 being non trivial RealLinearSpace holds ProjectiveSpace b1 is Vebleian
Lemma68:
for b1 being non trivial RealLinearSpace holds
( b1 is up-3-dimensional implies ProjectiveSpace b1 is proper )
theorem Th26: :: ANPROJ_2:26
Lemma70:
for b1 being non trivial RealLinearSpace holds
( b1 is up-3-dimensional implies ProjectiveSpace b1 is at_least_3rank )
Lemma71:
for b1 being up-3-dimensional RealLinearSpace holds
ProjectiveSpace b1 is CollProjectiveSpace
;
definition
let c
1 be
CollProjectiveSpace;
attr a
1 is
Fanoian means :
Def11:
:: ANPROJ_2:def 11
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
Element of a
1 holds
not ( b
1,b
2,b
3 is_collinear & b
4,b
5,b
3 is_collinear & b
1,b
4,b
6 is_collinear & b
2,b
5,b
6 is_collinear & b
1,b
5,b
7 is_collinear & b
2,b
4,b
7 is_collinear & b
6,b
3,b
7 is_collinear & not b
1,b
2,b
5 is_collinear & not b
1,b
2,b
4 is_collinear & not b
1,b
4,b
5 is_collinear & not b
2,b
4,b
5 is_collinear );
attr a
1 is
Desarguesian means :: ANPROJ_2:def 12
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( b
1 <> b
5 & b
2 <> b
5 & b
1 <> b
6 & b
3 <> b
6 & b
1 <> b
7 & b
4 <> b
7 & not b
1,b
2,b
3 is_collinear & not b
1,b
2,b
4 is_collinear & not b
1,b
3,b
4 is_collinear & b
2,b
3,b
10 is_collinear & b
5,b
6,b
10 is_collinear & b
3,b
4,b
8 is_collinear & b
6,b
7,b
8 is_collinear & b
2,b
4,b
9 is_collinear & b
5,b
7,b
9 is_collinear & b
1,b
2,b
5 is_collinear & b
1,b
3,b
6 is_collinear & b
1,b
4,b
7 is_collinear implies b
8,b
9,b
10 is_collinear );
attr a
1 is
Pappian means :: ANPROJ_2:def 13
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( b
1 <> b
3 & b
1 <> b
4 & b
3 <> b
4 & b
2 <> b
3 & b
2 <> b
4 & b
1 <> b
6 & b
1 <> b
7 & b
6 <> b
7 & b
5 <> b
6 & b
5 <> b
7 & not b
1,b
2,b
5 is_collinear & b
1,b
2,b
3 is_collinear & b
1,b
2,b
4 is_collinear & b
1,b
5,b
6 is_collinear & b
1,b
5,b
7 is_collinear & b
2,b
6,b
10 is_collinear & b
5,b
3,b
10 is_collinear & b
2,b
7,b
9 is_collinear & b
4,b
5,b
9 is_collinear & b
3,b
7,b
8 is_collinear & b
4,b
6,b
8 is_collinear implies b
8,b
9,b
10 is_collinear );
end;
:: deftheorem Def11 defines Fanoian ANPROJ_2:def 11 :
for b
1 being
CollProjectiveSpace holds
( b
1 is
Fanoian iff for 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
4 is_collinear & b
2,b
5,b
7 is_collinear & b
3,b
6,b
7 is_collinear & b
2,b
6,b
8 is_collinear & b
3,b
5,b
8 is_collinear & b
7,b
4,b
8 is_collinear & not b
2,b
3,b
6 is_collinear & not b
2,b
3,b
5 is_collinear & not b
2,b
5,b
6 is_collinear & not b
3,b
5,b
6 is_collinear ) );
:: deftheorem Def12 defines Desarguesian ANPROJ_2:def 12 :
for b
1 being
CollProjectiveSpace holds
( b
1 is
Desarguesian iff for 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
6 & b
3 <> b
6 & b
2 <> b
7 & b
4 <> b
7 & b
2 <> b
8 & b
5 <> b
8 & not b
2,b
3,b
4 is_collinear & not b
2,b
3,b
5 is_collinear & not b
2,b
4,b
5 is_collinear & b
3,b
4,b
11 is_collinear & b
6,b
7,b
11 is_collinear & b
4,b
5,b
9 is_collinear & b
7,b
8,b
9 is_collinear & b
3,b
5,b
10 is_collinear & b
6,b
8,b
10 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 implies b
9,b
10,b
11 is_collinear ) );
:: deftheorem Def13 defines Pappian ANPROJ_2:def 13 :
for b
1 being
CollProjectiveSpace holds
( b
1 is
Pappian iff for 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
4 & b
2 <> b
5 & b
4 <> b
5 & b
3 <> b
4 & b
3 <> b
5 & b
2 <> b
7 & b
2 <> b
8 & b
7 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & not b
2,b
3,b
6 is_collinear & b
2,b
3,b
4 is_collinear & b
2,b
3,b
5 is_collinear & b
2,b
6,b
7 is_collinear & b
2,b
6,b
8 is_collinear & b
3,b
7,b
11 is_collinear & b
6,b
4,b
11 is_collinear & b
3,b
8,b
10 is_collinear & b
5,b
6,b
10 is_collinear & b
4,b
8,b
9 is_collinear & b
5,b
7,b
9 is_collinear implies b
9,b
10,b
11 is_collinear ) );
:: deftheorem Def14 defines 2-dimensional ANPROJ_2:def 14 :
definition
let c
1 be
CollProjectiveSpace;
attr a
1 is
at_most-3-dimensional means :
Def15:
:: ANPROJ_2:def 15
for b
1, b
2, b
3, b
4, b
5 being
Element of a
1 holds
ex b
6, b
7 being
Element of a
1 st
( b
1,b
3,b
6 is_collinear & b
2,b
4,b
7 is_collinear & b
5,b
6,b
7 is_collinear );
end;
:: deftheorem Def15 defines at_most-3-dimensional ANPROJ_2:def 15 :
for b
1 being
CollProjectiveSpace holds
( b
1 is
at_most-3-dimensional iff for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
ex b
7, b
8 being
Element of b
1 st
( b
2,b
4,b
7 is_collinear & b
3,b
5,b
8 is_collinear & b
6,b
7,b
8 is_collinear ) );
theorem Th27: :: ANPROJ_2:27
canceled;
theorem Th28: :: ANPROJ_2:28
for b
1 being non
trivial RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Element of
(ProjectiveSpace b1) holds
not ( b
2,b
3,b
4 is_collinear & b
5,b
6,b
4 is_collinear & b
2,b
5,b
7 is_collinear & b
3,b
6,b
7 is_collinear & b
2,b
6,b
8 is_collinear & b
3,b
5,b
8 is_collinear & b
7,b
4,b
8 is_collinear & not b
2,b
3,b
6 is_collinear & not b
2,b
3,b
5 is_collinear & not b
2,b
5,b
6 is_collinear & not b
3,b
5,b
6 is_collinear )
Lemma76:
for b1 being up-3-dimensional RealLinearSpace holds ProjectiveSpace b1 is Fanoian
Lemma77:
for b1 being up-3-dimensional RealLinearSpace holds ProjectiveSpace b1 is Desarguesian
Lemma78:
for b1 being up-3-dimensional RealLinearSpace holds ProjectiveSpace b1 is Pappian
theorem Th29: :: ANPROJ_2:29
for b
1 being non
trivial RealLinearSpace holds
not ( ex b
2, b
3, b
4 being
Element of b
1 st
( ( for b
5, b
6, b
7 being
Real holds
(
((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b
1 implies ( b
5 = 0 & b
6 = 0 & b
7 = 0 ) ) ) & ( for b
5 being
Element of b
1 holds
ex b
6, b
7, b
8 being
Real st b
5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4) ) ) & ( for b
2, b
3 being
Element of
(ProjectiveSpace b1) holds
not ( b
2 <> b
3 & ( for b
4, b
5 being
Element of
(ProjectiveSpace b1) holds
ex b
6 being
Element of
(ProjectiveSpace b1) st
( b
2,b
3,b
6 is_collinear & b
4,b
5,b
6 is_collinear ) ) ) ) )
theorem Th30: :: ANPROJ_2:30
for b
1 being non
trivial RealLinearSpace holds
( ex b
2, b
3 being
Element of
(ProjectiveSpace b1) st
( b
2 <> b
3 & ( for b
4, b
5 being
Element of
(ProjectiveSpace b1) holds
ex b
6 being
Element of
(ProjectiveSpace b1) st
( b
2,b
3,b
6 is_collinear & b
4,b
5,b
6 is_collinear ) ) ) implies for b
2, b
3, b
4, b
5 being
Element of
(ProjectiveSpace b1) holds
ex b
6 being
Element of
(ProjectiveSpace b1) st
( b
2,b
3,b
6 is_collinear & b
4,b
5,b
6 is_collinear ) )
theorem Th31: :: ANPROJ_2:31
Lemma82:
for b1 being non trivial RealLinearSpace holds
( ex b2, b3, b4, b5 being Element of b1 st
for b6, b7, b8, b9 being Real holds
( (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1 implies ( b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0 ) ) implies b1 is up-3-dimensional )
Lemma83:
for b1 being non trivial RealLinearSpace holds
( ex b2, b3, b4, b5 being Element of b1 st
for b6, b7, b8, b9 being Real holds
( (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1 implies ( b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0 ) ) implies ( ProjectiveSpace b1 is proper & ProjectiveSpace b1 is at_least_3rank ) )
theorem Th32: :: ANPROJ_2:32
for b
1 being non
trivial RealLinearSpace holds
not ( ex b
2, b
3, b
4, b
5 being
Element of b
1 st
( ( for b
6 being
Element of b
1 holds
ex b
7, b
8, b
9, b
10 being
Real st b
6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5) ) & ( for b
6, b
7, b
8, b
9 being
Real holds
(
(((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b
1 implies ( b
6 = 0 & b
7 = 0 & b
8 = 0 & b
9 = 0 ) ) ) ) & ( for b
2, b
3, b
4 being
Element of
(ProjectiveSpace b1) holds
not ( not b
2,b
3,b
4 is_collinear & ( for b
5, b
6 being
Element of
(ProjectiveSpace b1) holds
ex b
7, b
8 being
Element of
(ProjectiveSpace b1) st
( b
5,b
6,b
8 is_collinear & b
3,b
4,b
7 is_collinear & b
2,b
8,b
7 is_collinear ) ) ) ) )
Lemma85:
for b1 being non trivial RealLinearSpace
for b2, b3, b4, b5, b6, b7 being Element of (ProjectiveSpace b1) holds
not ( not b7,b4,b3 is_collinear & b4,b3,b2 is_collinear & b7,b5,b4 is_collinear & b7,b6,b3 is_collinear & ( for b8 being Element of (ProjectiveSpace b1) holds
not ( b5,b6,b8 is_collinear & b7,b2,b8 is_collinear ) ) )
theorem Th33: :: ANPROJ_2:33
for b
1 being non
trivial RealLinearSpace holds
not (
ProjectiveSpace b
1 is
proper &
ProjectiveSpace b
1 is
at_least_3rank & ex b
2, b
3, b
4 being
Element of
(ProjectiveSpace b1) st
( not b
2,b
3,b
4 is_collinear & ( for b
5, b
6 being
Element of
(ProjectiveSpace b1) holds
ex b
7, b
8 being
Element of
(ProjectiveSpace b1) st
( b
5,b
6,b
8 is_collinear & b
3,b
4,b
7 is_collinear & b
2,b
8,b
7 is_collinear ) ) ) & ( for b
2 being
CollProjectiveSpace holds
not ( b
2 = ProjectiveSpace b
1 & b
2 is
at_most-3-dimensional ) ) )
theorem Th34: :: ANPROJ_2:34
theorem Th35: :: ANPROJ_2:35
theorem Th36: :: ANPROJ_2:36
theorem Th37: :: ANPROJ_2:37