Article ANPROJ_2, MML version 4.99.1005
:: ANPROJ_2:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st for b5, b6, b7 being Element of REAL
st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
holds b5 = 0 & b6 = 0 & b7 = 0
holds b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1) & not b2,b3,b4 are_LinDep & not are_Prop b2,b3;
:: ANPROJ_2:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0
holds b2 is non-zero(b1) & b3 is non-zero(b1) & not are_Prop b2,b3 & b4 is non-zero(b1) & b5 is non-zero(b1) & not are_Prop b4,b5 & not b2,b3,b4 are_LinDep & not b4,b5,b2 are_LinDep;
:: ANPROJ_2:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st (for b5 being Element of the carrier of b1 holds
ex b6, b7, b8 being Element of REAL st
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4)) &
(for b5, b6, b7 being Element of REAL
st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
holds b5 = 0 & b6 = 0 & b7 = 0)
for b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
b2,b3,b7 are_LinDep & b5,b6,b7 are_LinDep & b7 is non-zero(b1);
:: ANPROJ_2:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st (for b6 being Element of the carrier of b1 holds
ex b7, b8, b9, b10 being Element of REAL st
b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
(for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
for b6, b7 being Element of the carrier of b1
st b6 is non-zero(b1) & b7 is non-zero(b1)
holds ex b8, b9 being Element of the carrier of b1 st
b6,b7,b9 are_LinDep & b3,b4,b8 are_LinDep & b2,b9,b8 are_LinDep & b8 is non-zero(b1) & b9 is non-zero(b1);
:: ANPROJ_2:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0
for b6 being Element of the carrier of b1
st b6 is non-zero(b1) & b2,b3,b6 are_LinDep
holds not b4,b5,b6 are_LinDep;
:: ANPROJ_2:prednot 1 => ANPROJ_2:pred 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4 be Element of the carrier of a1;
pred A2,A3,A4 are_Prop_Vect means
a2 is non-zero(a1) & a3 is non-zero(a1) & a4 is non-zero(a1);
end;
:: ANPROJ_2:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4 be Element of the carrier of a1;
To prove
a2,a3,a4 are_Prop_Vect
it is sufficient to prove
thus a2 is non-zero(a1) & a3 is non-zero(a1) & a4 is non-zero(a1);
:: ANPROJ_2:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3,b4 are_Prop_Vect
iff
b2 is non-zero(b1) & b3 is non-zero(b1) & b4 is non-zero(b1);
:: ANPROJ_2:prednot 2 => ANPROJ_2:pred 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
pred A2,A3,A4,A5,A6,A7 lie_on_a_triangle means
a2,a3,a7 are_LinDep & a2,a4,a6 are_LinDep & a3,a4,a5 are_LinDep;
end;
:: ANPROJ_2:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
To prove
a2,a3,a4,a5,a6,a7 lie_on_a_triangle
it is sufficient to prove
thus a2,a3,a7 are_LinDep & a2,a4,a6 are_LinDep & a3,a4,a5 are_LinDep;
:: ANPROJ_2:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
b2,b3,b4,b5,b6,b7 lie_on_a_triangle
iff
b2,b3,b7 are_LinDep & b2,b4,b6 are_LinDep & b3,b4,b5 are_LinDep;
:: ANPROJ_2:prednot 3 => ANPROJ_2:pred 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
pred A2,A3,A4,A5,A6,A7,A8 are_perspective means
a2,a3,a6 are_LinDep & a2,a4,a7 are_LinDep & a2,a5,a8 are_LinDep;
end;
:: ANPROJ_2:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
To prove
a2,a3,a4,a5,a6,a7,a8 are_perspective
it is sufficient to prove
thus a2,a3,a6 are_LinDep & a2,a4,a7 are_LinDep & a2,a5,a8 are_LinDep;
:: ANPROJ_2:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1 holds
b2,b3,b4,b5,b6,b7,b8 are_perspective
iff
b2,b3,b6 are_LinDep & b2,b4,b7 are_LinDep & b2,b5,b8 are_LinDep;
:: ANPROJ_2:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3,b4 are_LinDep & not are_Prop b2,b3 & not are_Prop b2,b4 & not are_Prop b3,b4 & b2,b3,b4 are_Prop_Vect
holds (ex b5, b6 being Element of REAL st
b6 * b4 = b2 + (b5 * b3) & b5 <> 0 & b6 <> 0) &
(ex b5, b6 being Element of REAL st
b4 = (b6 * b2) + (b5 * b3) & b6 <> 0 & b5 <> 0);
:: ANPROJ_2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st b2,b3,b4 are_LinDep & not are_Prop b2,b3 & b2,b3,b4 are_Prop_Vect
holds ex b5, b6 being Element of REAL st
b4 = (b5 * b2) + (b6 * b3);
:: ANPROJ_2:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b2 is non-zero(b1) & b3,b4,b5 are_Prop_Vect & b6,b7,b8 are_Prop_Vect & b9,b10,b11 are_Prop_Vect & b2,b3,b4,b5,b6,b7,b8 are_perspective & not are_Prop b2,b6 & not are_Prop b2,b7 & not are_Prop b2,b8 & not are_Prop b3,b6 & not are_Prop b4,b7 & not are_Prop b5,b8 & not b2,b3,b4 are_LinDep & not b2,b3,b5 are_LinDep & not b2,b4,b5 are_LinDep & b3,b4,b5,b9,b10,b11 lie_on_a_triangle & b6,b7,b8,b9,b10,b11 lie_on_a_triangle
holds b9,b10,b11 are_LinDep;
:: ANPROJ_2:prednot 4 => ANPROJ_2:pred 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
pred A2,A3,A4,A5,A6,A7,A8 lie_on_an_angle means
not a2,a3,a6 are_LinDep & a2,a3,a4 are_LinDep & a2,a3,a5 are_LinDep & a2,a6,a7 are_LinDep & a2,a6,a8 are_LinDep;
end;
:: ANPROJ_2:dfs 4
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
To prove
a2,a3,a4,a5,a6,a7,a8 lie_on_an_angle
it is sufficient to prove
thus not a2,a3,a6 are_LinDep & a2,a3,a4 are_LinDep & a2,a3,a5 are_LinDep & a2,a6,a7 are_LinDep & a2,a6,a8 are_LinDep;
:: ANPROJ_2:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1 holds
b2,b3,b4,b5,b6,b7,b8 lie_on_an_angle
iff
not b2,b3,b6 are_LinDep & b2,b3,b4 are_LinDep & b2,b3,b5 are_LinDep & b2,b6,b7 are_LinDep & b2,b6,b8 are_LinDep;
:: ANPROJ_2:prednot 5 => ANPROJ_2:pred 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
pred A2,A3,A4,A5,A6,A7,A8 are_half_mutually_not_Prop means
not are_Prop a2,a4 & not are_Prop a2,a5 & not are_Prop a2,a7 & not are_Prop a2,a8 & not are_Prop a3,a4 & not are_Prop a3,a5 & not are_Prop a6,a7 & not are_Prop a6,a8 & not are_Prop a4,a5 & not are_Prop a7,a8;
end;
:: ANPROJ_2:dfs 5
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7, a8 be Element of the carrier of a1;
To prove
a2,a3,a4,a5,a6,a7,a8 are_half_mutually_not_Prop
it is sufficient to prove
thus not are_Prop a2,a4 & not are_Prop a2,a5 & not are_Prop a2,a7 & not are_Prop a2,a8 & not are_Prop a3,a4 & not are_Prop a3,a5 & not are_Prop a6,a7 & not are_Prop a6,a8 & not are_Prop a4,a5 & not are_Prop a7,a8;
:: ANPROJ_2:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1 holds
b2,b3,b4,b5,b6,b7,b8 are_half_mutually_not_Prop
iff
not are_Prop b2,b4 & not are_Prop b2,b5 & not are_Prop b2,b7 & not are_Prop b2,b8 & not are_Prop b3,b4 & not are_Prop b3,b5 & not are_Prop b6,b7 & not are_Prop b6,b8 & not are_Prop b4,b5 & not are_Prop b7,b8;
:: ANPROJ_2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b2 is non-zero(b1) & b3,b4,b5 are_Prop_Vect & b6,b7,b8 are_Prop_Vect & b9,b10,b11 are_Prop_Vect & b2,b3,b4,b5,b6,b7,b8 lie_on_an_angle & b2,b3,b4,b5,b6,b7,b8 are_half_mutually_not_Prop & b3,b7,b11 are_LinDep & b6,b4,b11 are_LinDep & b3,b8,b10 are_LinDep & b5,b6,b10 are_LinDep & b4,b8,b9 are_LinDep & b5,b7,b9 are_LinDep
holds b9,b10,b11 are_LinDep;
:: ANPROJ_2:th 10
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1 holds
ex b5, b6, b7 being Element of Funcs(b1,REAL) st
(for b8 being set
st b8 in b1
holds (b8 = b2 implies b5 . b8 = 1) & (b8 = b2 or b5 . b8 = 0)) &
(for b8 being set
st b8 in b1
holds (b8 = b3 implies b6 . b8 = 1) & (b8 = b3 or b6 . b8 = 0)) &
(for b8 being set
st b8 in b1
holds (b8 = b4 implies b7 . b8 = 1) & (b8 = b4 or b7 . b8 = 0));
:: ANPROJ_2:th 11
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,REAL)
for b5, b6, b7 being Element of b1
st b5 in b1 &
b6 in b1 &
b7 in b1 &
b5 <> b6 &
b5 <> b7 &
b6 <> b7 &
(for b8 being set
st b8 in b1
holds (b8 = b5 implies b2 . b8 = 1) & (b8 = b5 or b2 . b8 = 0)) &
(for b8 being set
st b8 in b1
holds (b8 = b6 implies b3 . b8 = 1) & (b8 = b6 or b3 . b8 = 0)) &
(for b8 being set
st b8 in b1
holds (b8 = b7 implies b4 . b8 = 1) & (b8 = b7 or b4 . b8 = 0))
for b8, b9, b10 being Element of REAL
st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b8,b2],(RealFuncExtMult b1) . [b9,b3]),(RealFuncExtMult b1) . [b10,b4]) = RealFuncZero b1
holds b8 = 0 & b9 = 0 & b10 = 0;
:: ANPROJ_2:th 12
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
st b2 in b1 & b3 in b1 & b4 in b1 & b2 <> b3 & b2 <> b4 & b3 <> b4
holds ex b5, b6, b7 being Element of Funcs(b1,REAL) st
for b8, b9, b10 being Element of REAL
st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b8,b5],(RealFuncExtMult b1) . [b9,b6]),(RealFuncExtMult b1) . [b10,b7]) = RealFuncZero b1
holds b8 = 0 & b9 = 0 & b10 = 0;
:: ANPROJ_2:th 13
theorem
for b1 being non empty set
for b2, b3, b4 being Element of Funcs(b1,REAL)
for b5, b6, b7 being Element of b1
st b1 = {b5,b6,b7} &
b5 <> b6 &
b5 <> b7 &
b6 <> b7 &
(for b8 being set
st b8 in b1
holds (b8 = b5 implies b2 . b8 = 1) & (b8 = b5 or b2 . b8 = 0)) &
(for b8 being set
st b8 in b1
holds (b8 = b6 implies b3 . b8 = 1) & (b8 = b6 or b3 . b8 = 0)) &
(for b8 being set
st b8 in b1
holds (b8 = b7 implies b4 . b8 = 1) & (b8 = b7 or b4 . b8 = 0))
for b8 being Element of Funcs(b1,REAL) holds
ex b9, b10, b11 being Element of REAL st
b8 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b9,b2],(RealFuncExtMult b1) . [b10,b3]),(RealFuncExtMult b1) . [b11,b4]);
:: ANPROJ_2:th 14
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
st b1 = {b2,b3,b4} & b2 <> b3 & b2 <> b4 & b3 <> b4
holds ex b5, b6, b7 being Element of Funcs(b1,REAL) st
for b8 being Element of Funcs(b1,REAL) holds
ex b9, b10, b11 being Element of REAL st
b8 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b9,b5],(RealFuncExtMult b1) . [b10,b6]),(RealFuncExtMult b1) . [b11,b7]);
:: ANPROJ_2:th 15
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
st b1 = {b2,b3,b4} & b2 <> b3 & b2 <> b4 & b3 <> b4
holds ex b5, b6, b7 being Element of Funcs(b1,REAL) st
(for b8, b9, b10 being Element of REAL
st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b8,b5],(RealFuncExtMult b1) . [b9,b6]),(RealFuncExtMult b1) . [b10,b7]) = RealFuncZero b1
holds b8 = 0 & b9 = 0 & b10 = 0) &
(for b8 being Element of Funcs(b1,REAL) holds
ex b9, b10, b11 being Element of REAL st
b8 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b9,b5],(RealFuncExtMult b1) . [b10,b6]),(RealFuncExtMult b1) . [b11,b7]));
:: ANPROJ_2:th 16
theorem
ex b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct st
ex b2, b3, b4 being Element of the carrier of b1 st
(for b5, b6, b7 being Element of REAL
st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
holds b5 = 0 & b6 = 0 & b7 = 0) &
(for b5 being Element of the carrier of b1 holds
ex b6, b7, b8 being Element of REAL st
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4));
:: ANPROJ_2:th 17
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of b1 holds
ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
(for b10 being set
st b10 in b1
holds (b10 = b2 implies b6 . b10 = 1) & (b10 = b2 or b6 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b3 implies b7 . b10 = 1) & (b10 = b3 or b7 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b4 implies b8 . b10 = 1) & (b10 = b4 or b8 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b5 implies b9 . b10 = 1) & (b10 = b5 or b9 . b10 = 0));
:: ANPROJ_2:th 18
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,REAL)
for b6, b7, b8, b9 being Element of b1
st b6 in b1 &
b7 in b1 &
b8 in b1 &
b9 in b1 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 &
(for b10 being set
st b10 in b1
holds (b10 = b6 implies b2 . b10 = 1) & (b10 = b6 or b2 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b7 implies b3 . b10 = 1) & (b10 = b7 or b3 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b8 implies b4 . b10 = 1) & (b10 = b8 or b4 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b9 implies b5 . b10 = 1) & (b10 = b9 or b5 . b10 = 0))
for b10, b11, b12, b13 being Element of REAL
st (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b10,b2],(RealFuncExtMult b1) . [b11,b3]),(RealFuncExtMult b1) . [b12,b4]),(RealFuncExtMult b1) . [b13,b5]) = RealFuncZero b1
holds b10 = 0 & b11 = 0 & b12 = 0 & b13 = 0;
:: ANPROJ_2:th 19
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of b1
st b2 in b1 & b3 in b1 & b4 in b1 & b5 in b1 & b2 <> b3 & b2 <> b4 & b2 <> b5 & b3 <> b4 & b3 <> b5 & b4 <> b5
holds ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
for b10, b11, b12, b13 being Element of 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 b1
holds b10 = 0 & b11 = 0 & b12 = 0 & b13 = 0;
:: ANPROJ_2:th 20
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of Funcs(b1,REAL)
for b6, b7, b8, b9 being Element of b1
st b1 = {b6,b7,b8,b9} &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 &
(for b10 being set
st b10 in b1
holds (b10 = b6 implies b2 . b10 = 1) & (b10 = b6 or b2 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b7 implies b3 . b10 = 1) & (b10 = b7 or b3 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b8 implies b4 . b10 = 1) & (b10 = b8 or b4 . b10 = 0)) &
(for b10 being set
st b10 in b1
holds (b10 = b9 implies b5 . b10 = 1) & (b10 = b9 or b5 . b10 = 0))
for b10 being Element of Funcs(b1,REAL) holds
ex b11, b12, b13, b14 being Element of REAL st
b10 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b11,b2],(RealFuncExtMult b1) . [b12,b3]),(RealFuncExtMult b1) . [b13,b4]),(RealFuncExtMult b1) . [b14,b5]);
:: ANPROJ_2:th 21
theorem
for b1 being non empty set
for 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
holds ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
for b10 being Element of Funcs(b1,REAL) holds
ex b11, b12, b13, b14 being Element of REAL st
b10 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b11,b6],(RealFuncExtMult b1) . [b12,b7]),(RealFuncExtMult b1) . [b13,b8]),(RealFuncExtMult b1) . [b14,b9]);
:: ANPROJ_2:th 22
theorem
for b1 being non empty set
for 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
holds ex b6, b7, b8, b9 being Element of Funcs(b1,REAL) st
(for b10, b11, b12, b13 being Element of 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 b1
holds b10 = 0 & b11 = 0 & b12 = 0 & b13 = 0) &
(for b10 being Element of Funcs(b1,REAL) holds
ex b11, b12, b13, b14 being Element of REAL st
b10 = (RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncAdd b1) .((RealFuncExtMult b1) . [b11,b6],(RealFuncExtMult b1) . [b12,b7]),(RealFuncExtMult b1) . [b13,b8]),(RealFuncExtMult b1) . [b14,b9]));
:: ANPROJ_2:th 23
theorem
ex b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct st
ex b2, b3, b4, b5 being Element of the carrier of b1 st
(for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0) &
(for b6 being Element of the carrier of b1 holds
ex b7, b8, b9, b10 being Element of REAL st
b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5));
:: ANPROJ_2:attrnot 1 => ANPROJ_2:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
attr a1 is up-3-dimensional means
ex b1, b2, b3 being Element of the carrier of a1 st
for b4, b5, b6 being Element of REAL
st ((b4 * b1) + (b5 * b2)) + (b6 * b3) = 0. a1
holds b4 = 0 & b5 = 0 & b6 = 0;
end;
:: ANPROJ_2:dfs 6
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
To prove
a1 is up-3-dimensional
it is sufficient to prove
thus ex b1, b2, b3 being Element of the carrier of a1 st
for b4, b5, b6 being Element of REAL
st ((b4 * b1) + (b5 * b2)) + (b6 * b3) = 0. a1
holds b4 = 0 & b5 = 0 & b6 = 0;
:: ANPROJ_2:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
b1 is up-3-dimensional
iff
ex b2, b3, b4 being Element of the carrier of b1 st
for b5, b6, b7 being Element of REAL
st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
holds b5 = 0 & b6 = 0 & b7 = 0;
:: ANPROJ_2:exreg 1
registration
cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
end;
:: ANPROJ_2:condreg 1
registration
cluster non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional -> non trivial (RLSStruct);
end;
:: ANPROJ_2:attrnot 2 => COLLSP:attr 2
definition
let a1 be non empty CollStr;
attr a1 is reflexive means
for b1, b2, b3 being Element of the carrier of a1 holds
b1,b2,b1 is_collinear & b1,b1,b2 is_collinear & b1,b2,b2 is_collinear;
end;
:: ANPROJ_2:dfs 7
definiens
let a1 be non empty CollStr;
To prove
a1 is reflexive
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
b1,b2,b1 is_collinear & b1,b1,b2 is_collinear & b1,b2,b2 is_collinear;
:: ANPROJ_2:def 7
theorem
for b1 being non empty CollStr holds
b1 is reflexive
iff
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3,b2 is_collinear & b2,b2,b3 is_collinear & b2,b3,b3 is_collinear;
:: ANPROJ_2:attrnot 3 => COLLSP:attr 3
definition
let a1 be non empty CollStr;
attr a1 is transitive means
for b1, b2, b3, b4, b5 being Element of the carrier of a1
st b1 <> b2 & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b2,b5 is_collinear
holds b3,b4,b5 is_collinear;
end;
:: ANPROJ_2:dfs 8
definiens
let a1 be non empty CollStr;
To prove
a1 is transitive
it is sufficient to prove
thus for b1, b2, b3, b4, b5 being Element of the carrier of a1
st b1 <> b2 & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b2,b5 is_collinear
holds b3,b4,b5 is_collinear;
:: ANPROJ_2:def 8
theorem
for b1 being non empty CollStr holds
b1 is transitive
iff
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2 <> b3 & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b3,b6 is_collinear
holds b4,b5,b6 is_collinear;
:: ANPROJ_2:attrnot 4 => ANPROJ_2:attr 2
definition
let a1 be non empty CollStr;
attr a1 is Vebleian means
for b1, b2, b3, b4, b5 being Element of the carrier of a1
st b1,b2,b4 is_collinear & b2,b3,b5 is_collinear
holds ex b6 being Element of the carrier of a1 st
b1,b3,b6 is_collinear & b4,b5,b6 is_collinear;
end;
:: ANPROJ_2:dfs 9
definiens
let a1 be non empty CollStr;
To prove
a1 is Vebleian
it is sufficient to prove
thus for b1, b2, b3, b4, b5 being Element of the carrier of a1
st b1,b2,b4 is_collinear & b2,b3,b5 is_collinear
holds ex b6 being Element of the carrier of a1 st
b1,b3,b6 is_collinear & b4,b5,b6 is_collinear;
:: ANPROJ_2:def 9
theorem
for b1 being non empty CollStr holds
b1 is Vebleian
iff
for b2, b3, b4, b5, b6 being Element of the carrier of b1
st b2,b3,b5 is_collinear & b3,b4,b6 is_collinear
holds ex b7 being Element of the carrier of b1 st
b2,b4,b7 is_collinear & b5,b6,b7 is_collinear;
:: ANPROJ_2:attrnot 5 => ANPROJ_2:attr 3
definition
let a1 be non empty CollStr;
attr a1 is at_least_3rank means
for b1, b2 being Element of the carrier of a1 holds
ex b3 being Element of the carrier of a1 st
b1 <> b3 & b2 <> b3 & b1,b2,b3 is_collinear;
end;
:: ANPROJ_2:dfs 10
definiens
let a1 be non empty CollStr;
To prove
a1 is at_least_3rank
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
ex b3 being Element of the carrier of a1 st
b1 <> b3 & b2 <> b3 & b1,b2,b3 is_collinear;
:: ANPROJ_2:def 10
theorem
for b1 being non empty CollStr holds
b1 is at_least_3rank
iff
for b2, b3 being Element of the carrier of b1 holds
ex b4 being Element of the carrier of b1 st
b2 <> b4 & b3 <> b4 & b2,b3,b4 is_collinear;
:: ANPROJ_2:th 24
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of ProjectiveSpace b1 holds
b2,b3,b4 is_collinear
iff
ex b5, b6, b7 being Element of the carrier of b1 st
b2 = Dir b5 & b3 = Dir b6 & b4 = Dir b7 & b5 is non-zero(b1) & b6 is non-zero(b1) & b7 is non-zero(b1) & b5,b6,b7 are_LinDep;
:: ANPROJ_2:funcreg 1
registration
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster ProjectiveSpace a1 -> strict reflexive transitive;
end;
:: ANPROJ_2:th 25
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of ProjectiveSpace b1
st b2,b3,b4 is_collinear
holds b2,b4,b3 is_collinear & b3,b2,b4 is_collinear & b4,b3,b2 is_collinear & b4,b2,b3 is_collinear & b3,b4,b2 is_collinear;
:: ANPROJ_2:funcreg 2
registration
let a1 be non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster ProjectiveSpace a1 -> strict Vebleian;
end;
:: ANPROJ_2:funcreg 3
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
cluster ProjectiveSpace a1 -> strict proper;
end;
:: ANPROJ_2:th 26
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3 being Element of the carrier of b1 st
for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0
holds ProjectiveSpace b1 is at_least_3rank;
:: ANPROJ_2:funcreg 4
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
cluster ProjectiveSpace a1 -> strict at_least_3rank;
end;
:: ANPROJ_2:exreg 2
registration
cluster non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
end;
:: ANPROJ_2:modenot 1
definition
mode CollProjectiveSpace is non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
end;
:: ANPROJ_2:attrnot 6 => ANPROJ_2:attr 4
definition
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
attr a1 is Fanoian means
for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
st b1,b2,b3 is_collinear & b4,b5,b3 is_collinear & b1,b4,b6 is_collinear & b2,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b4,b7 is_collinear & b6,b3,b7 is_collinear & not b1,b2,b5 is_collinear & not b1,b2,b4 is_collinear & not b1,b4,b5 is_collinear
holds b2,b4,b5 is_collinear;
end;
:: ANPROJ_2:dfs 11
definiens
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
a1 is Fanoian
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
st b1,b2,b3 is_collinear & b4,b5,b3 is_collinear & b1,b4,b6 is_collinear & b2,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b4,b7 is_collinear & b6,b3,b7 is_collinear & not b1,b2,b5 is_collinear & not b1,b2,b4 is_collinear & not b1,b4,b5 is_collinear
holds b2,b4,b5 is_collinear;
:: ANPROJ_2:def 11
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
b1 is Fanoian
iff
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st b2,b3,b4 is_collinear & b5,b6,b4 is_collinear & b2,b5,b7 is_collinear & b3,b6,b7 is_collinear & b2,b6,b8 is_collinear & b3,b5,b8 is_collinear & b7,b4,b8 is_collinear & not b2,b3,b6 is_collinear & not b2,b3,b5 is_collinear & not b2,b5,b6 is_collinear
holds b3,b5,b6 is_collinear;
:: ANPROJ_2:attrnot 7 => ANPROJ_2:attr 5
definition
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
attr a1 is Desarguesian means
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b1 <> b5 & b2 <> b5 & b1 <> b6 & b3 <> b6 & b1 <> b7 & b4 <> b7 & not b1,b2,b3 is_collinear & not b1,b2,b4 is_collinear & not b1,b3,b4 is_collinear & b2,b3,b10 is_collinear & b5,b6,b10 is_collinear & b3,b4,b8 is_collinear & b6,b7,b8 is_collinear & b2,b4,b9 is_collinear & b5,b7,b9 is_collinear & b1,b2,b5 is_collinear & b1,b3,b6 is_collinear & b1,b4,b7 is_collinear
holds b8,b9,b10 is_collinear;
end;
:: ANPROJ_2:dfs 12
definiens
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
a1 is Desarguesian
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b1 <> b5 & b2 <> b5 & b1 <> b6 & b3 <> b6 & b1 <> b7 & b4 <> b7 & not b1,b2,b3 is_collinear & not b1,b2,b4 is_collinear & not b1,b3,b4 is_collinear & b2,b3,b10 is_collinear & b5,b6,b10 is_collinear & b3,b4,b8 is_collinear & b6,b7,b8 is_collinear & b2,b4,b9 is_collinear & b5,b7,b9 is_collinear & b1,b2,b5 is_collinear & b1,b3,b6 is_collinear & b1,b4,b7 is_collinear
holds b8,b9,b10 is_collinear;
:: ANPROJ_2:def 12
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
b1 is Desarguesian
iff
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b2 <> b6 & b3 <> b6 & b2 <> b7 & b4 <> b7 & b2 <> b8 & b5 <> b8 & not b2,b3,b4 is_collinear & not b2,b3,b5 is_collinear & not b2,b4,b5 is_collinear & b3,b4,b11 is_collinear & b6,b7,b11 is_collinear & b4,b5,b9 is_collinear & b7,b8,b9 is_collinear & b3,b5,b10 is_collinear & b6,b8,b10 is_collinear & b2,b3,b6 is_collinear & b2,b4,b7 is_collinear & b2,b5,b8 is_collinear
holds b9,b10,b11 is_collinear;
:: ANPROJ_2:attrnot 8 => ANPROJ_2:attr 6
definition
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
attr a1 is Pappian means
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b1 <> b3 & b1 <> b4 & b3 <> b4 & b2 <> b3 & b2 <> b4 & b1 <> b6 & b1 <> b7 & b6 <> b7 & b5 <> b6 & b5 <> b7 & not b1,b2,b5 is_collinear & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b6,b10 is_collinear & b5,b3,b10 is_collinear & b2,b7,b9 is_collinear & b4,b5,b9 is_collinear & b3,b7,b8 is_collinear & b4,b6,b8 is_collinear
holds b8,b9,b10 is_collinear;
end;
:: ANPROJ_2:dfs 13
definiens
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
a1 is Pappian
it is sufficient to prove
thus for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10 being Element of the carrier of a1
st b1 <> b3 & b1 <> b4 & b3 <> b4 & b2 <> b3 & b2 <> b4 & b1 <> b6 & b1 <> b7 & b6 <> b7 & b5 <> b6 & b5 <> b7 & not b1,b2,b5 is_collinear & b1,b2,b3 is_collinear & b1,b2,b4 is_collinear & b1,b5,b6 is_collinear & b1,b5,b7 is_collinear & b2,b6,b10 is_collinear & b5,b3,b10 is_collinear & b2,b7,b9 is_collinear & b4,b5,b9 is_collinear & b3,b7,b8 is_collinear & b4,b6,b8 is_collinear
holds b8,b9,b10 is_collinear;
:: ANPROJ_2:def 13
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
b1 is Pappian
iff
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st b2 <> b4 & b2 <> b5 & b4 <> b5 & b3 <> b4 & b3 <> b5 & b2 <> b7 & b2 <> b8 & b7 <> b8 & b6 <> b7 & b6 <> b8 & not b2,b3,b6 is_collinear & b2,b3,b4 is_collinear & b2,b3,b5 is_collinear & b2,b6,b7 is_collinear & b2,b6,b8 is_collinear & b3,b7,b11 is_collinear & b6,b4,b11 is_collinear & b3,b8,b10 is_collinear & b5,b6,b10 is_collinear & b4,b8,b9 is_collinear & b5,b7,b9 is_collinear
holds b9,b10,b11 is_collinear;
:: ANPROJ_2:attrnot 9 => ANPROJ_2:attr 7
definition
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
attr a1 is 2-dimensional means
for b1, b2, b3, b4 being Element of the carrier of a1 holds
ex b5 being Element of the carrier of a1 st
b1,b2,b5 is_collinear & b3,b4,b5 is_collinear;
end;
:: ANPROJ_2:dfs 14
definiens
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
a1 is 2-dimensional
it is sufficient to prove
thus for b1, b2, b3, b4 being Element of the carrier of a1 holds
ex b5 being Element of the carrier of a1 st
b1,b2,b5 is_collinear & b3,b4,b5 is_collinear;
:: ANPROJ_2:def 14
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
b1 is 2-dimensional
iff
for b2, b3, b4, b5 being Element of the carrier of b1 holds
ex b6 being Element of the carrier of b1 st
b2,b3,b6 is_collinear & b4,b5,b6 is_collinear;
:: ANPROJ_2:attrnot 10 => ANPROJ_2:attr 7
notation
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
antonym up-3-dimensional for 2-dimensional;
end;
:: ANPROJ_2:attrnot 11 => ANPROJ_2:attr 8
definition
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
attr a1 is at_most-3-dimensional means
for b1, b2, b3, b4, b5 being Element of the carrier of a1 holds
ex b6, b7 being Element of the carrier of a1 st
b1,b3,b6 is_collinear & b2,b4,b7 is_collinear & b5,b6,b7 is_collinear;
end;
:: ANPROJ_2:dfs 15
definiens
let a1 be non empty reflexive transitive proper Vebleian at_least_3rank CollStr;
To prove
a1 is at_most-3-dimensional
it is sufficient to prove
thus for b1, b2, b3, b4, b5 being Element of the carrier of a1 holds
ex b6, b7 being Element of the carrier of a1 st
b1,b3,b6 is_collinear & b2,b4,b7 is_collinear & b5,b6,b7 is_collinear;
:: ANPROJ_2:def 15
theorem
for b1 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr holds
b1 is at_most-3-dimensional
iff
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
ex b7, b8 being Element of the carrier of b1 st
b2,b4,b7 is_collinear & b3,b5,b8 is_collinear & b6,b7,b8 is_collinear;
:: ANPROJ_2:th 28
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of ProjectiveSpace b1
st b2,b3,b4 is_collinear & b5,b6,b4 is_collinear & b2,b5,b7 is_collinear & b3,b6,b7 is_collinear & b2,b6,b8 is_collinear & b3,b5,b8 is_collinear & b7,b4,b8 is_collinear & not b2,b3,b6 is_collinear & not b2,b3,b5 is_collinear & not b2,b5,b6 is_collinear
holds b3,b5,b6 is_collinear;
:: ANPROJ_2:funcreg 5
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like up-3-dimensional RLSStruct;
cluster ProjectiveSpace a1 -> strict Fanoian Desarguesian Pappian;
end;
:: ANPROJ_2:th 29
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3, b4 being Element of the carrier of b1 st
(for b5, b6, b7 being Element of REAL
st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
holds b5 = 0 & b6 = 0 & b7 = 0) &
(for b5 being Element of the carrier of b1 holds
ex b6, b7, b8 being Element of REAL st
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4))
holds ex b2, b3 being Element of the carrier of ProjectiveSpace b1 st
b2 <> b3 &
(for b4, b5 being Element of the carrier of ProjectiveSpace b1 holds
ex b6 being Element of the carrier of ProjectiveSpace b1 st
b2,b3,b6 is_collinear & b4,b5,b6 is_collinear);
:: ANPROJ_2:th 30
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3 being Element of the carrier of ProjectiveSpace b1 st
b2 <> b3 &
(for b4, b5 being Element of the carrier of ProjectiveSpace b1 holds
ex b6 being Element of the carrier of ProjectiveSpace b1 st
b2,b3,b6 is_collinear & b4,b5,b6 is_collinear)
for b2, b3, b4, b5 being Element of the carrier of ProjectiveSpace b1 holds
ex b6 being Element of the carrier of ProjectiveSpace b1 st
b2,b3,b6 is_collinear & b4,b5,b6 is_collinear;
:: ANPROJ_2:th 31
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3, b4 being Element of the carrier of b1 st
(for b5, b6, b7 being Element of REAL
st ((b5 * b2) + (b6 * b3)) + (b7 * b4) = 0. b1
holds b5 = 0 & b6 = 0 & b7 = 0) &
(for b5 being Element of the carrier of b1 holds
ex b6, b7, b8 being Element of REAL st
b5 = ((b6 * b2) + (b7 * b3)) + (b8 * b4))
holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
b2 = ProjectiveSpace b1 & b2 is 2-dimensional;
:: ANPROJ_2:th 32
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3, b4, b5 being Element of the carrier of b1 st
(for b6 being Element of the carrier of b1 holds
ex b7, b8, b9, b10 being Element of REAL st
b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
(for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
holds ex b2, b3, b4 being Element of the carrier of ProjectiveSpace b1 st
not b2,b3,b4 is_collinear &
(for b5, b6 being Element of the carrier of ProjectiveSpace b1 holds
ex b7, b8 being Element of the carrier of ProjectiveSpace b1 st
b5,b6,b8 is_collinear & b3,b4,b7 is_collinear & b2,b8,b7 is_collinear);
:: ANPROJ_2:th 33
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ProjectiveSpace b1 is proper &
ProjectiveSpace b1 is at_least_3rank &
(ex b2, b3, b4 being Element of the carrier of ProjectiveSpace b1 st
not b2,b3,b4 is_collinear &
(for b5, b6 being Element of the carrier of ProjectiveSpace b1 holds
ex b7, b8 being Element of the carrier of ProjectiveSpace b1 st
b5,b6,b8 is_collinear & b3,b4,b7 is_collinear & b2,b8,b7 is_collinear))
holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
b2 = ProjectiveSpace b1 & b2 is at_most-3-dimensional;
:: ANPROJ_2:th 34
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3, b4, b5 being Element of the carrier of b1 st
(for b6 being Element of the carrier of b1 holds
ex b7, b8, b9, b10 being Element of REAL st
b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
(for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
b2 = ProjectiveSpace b1 & b2 is at_most-3-dimensional;
:: ANPROJ_2:th 35
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3, b4, b5 being Element of the carrier of b1 st
for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0
holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
b2 = ProjectiveSpace b1 & b2 is up-3-dimensional;
:: ANPROJ_2:th 36
theorem
for b1 being non empty non trivial right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
st ex b2, b3, b4, b5 being Element of the carrier of b1 st
(for b6 being Element of the carrier of b1 holds
ex b7, b8, b9, b10 being Element of REAL st
b6 = (((b7 * b2) + (b8 * b3)) + (b9 * b4)) + (b10 * b5)) &
(for b6, b7, b8, b9 being Element of REAL
st (((b6 * b2) + (b7 * b3)) + (b8 * b4)) + (b9 * b5) = 0. b1
holds b6 = 0 & b7 = 0 & b8 = 0 & b9 = 0)
holds ex b2 being non empty reflexive transitive proper Vebleian at_least_3rank CollStr st
b2 = ProjectiveSpace b1 & b2 is up-3-dimensional & b2 is at_most-3-dimensional;
:: ANPROJ_2:exreg 3
registration
cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian 2-dimensional CollStr;
end;
:: ANPROJ_2:exreg 4
registration
cluster non empty strict reflexive transitive proper Vebleian at_least_3rank Fanoian Desarguesian Pappian up-3-dimensional at_most-3-dimensional CollStr;
end;
:: ANPROJ_2:modenot 2
definition
mode CollProjectivePlane is non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional CollStr;
end;
:: ANPROJ_2:th 37
theorem
for b1 being non empty CollStr holds
b1 is non empty reflexive transitive proper Vebleian at_least_3rank 2-dimensional CollStr
iff
b1 is non empty reflexive transitive proper at_least_3rank CollStr &
(for b2, b3, b4, b5 being Element of the carrier of b1 holds
ex b6 being Element of the carrier of b1 st
b2,b3,b6 is_collinear & b4,b5,b6 is_collinear);