Article PENCIL_4, MML version 4.99.1005
:: PENCIL_4:th 1
theorem
for b1, b2 being natural set
st 1 <= b1 & b1 < b2 & 3 <= b2 & b2 <= b1 + 1
holds 2 <= b1;
:: PENCIL_4:th 2
theorem
for b1 being finite set
for b2 being natural set
st b2 <= card b1
holds ex b3 being Element of bool b1 st
card b3 = b2;
:: PENCIL_4:th 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
b3 is Subspace of (Omega). b2;
:: PENCIL_4:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of (Omega). b2 holds
b3 is Subspace of b2;
:: PENCIL_4:th 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
(Omega). b3 is Subspace of b2;
:: PENCIL_4:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
st (Omega). b3 is Subspace of b2
holds b3 is Subspace of b2;
:: PENCIL_4:funcnot 1 => PENCIL_4:func 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3, a4 be Subspace of a2;
func segment(A3,A4) -> Element of bool Subspaces a2 means
for b1 being strict Subspace of a2 holds
b1 in it
iff
a3 is Subspace of b1 & b1 is Subspace of a4
if a3 is Subspace of a4
otherwise it = {};
end;
:: PENCIL_4:def 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
for b5 being Element of bool Subspaces b2 holds
(b3 is Subspace of b4 implies (b5 = segment(b3,b4)
iff
for b6 being strict Subspace of b2 holds
b6 in b5
iff
b3 is Subspace of b6 & b6 is Subspace of b4)) &
(b3 is Subspace of b4 or (b5 = segment(b3,b4)
iff
b5 = {}));
:: PENCIL_4:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5, b6 being Subspace of b2
st b3 is Subspace of b4 & b5 is Subspace of b6 & (Omega). b3 = (Omega). b5 & (Omega). b4 = (Omega). b6
holds segment(b3,b4) = segment(b5,b6);
:: PENCIL_4:funcnot 2 => PENCIL_4:func 2
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
let a3, a4 be Subspace of a2;
func pencil(A3,A4) -> Element of bool Subspaces a2 equals
(segment(a3,a4)) \ {(Omega). a3,(Omega). a4};
end;
:: PENCIL_4:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
pencil(b3,b4) = (segment(b3,b4)) \ {(Omega). b3,(Omega). b4};
:: PENCIL_4:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5, b6 being Subspace of b2
st b3 is Subspace of b4 & b5 is Subspace of b6 & (Omega). b3 = (Omega). b5 & (Omega). b4 = (Omega). b6
holds pencil(b3,b4) = pencil(b5,b6);
:: PENCIL_4:funcnot 3 => PENCIL_4:func 3
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3, a4 be Subspace of a2;
let a5 be natural set;
func pencil(A3,A4,A5) -> Element of bool (a5 Subspaces_of a2) equals
(pencil(a3,a4)) /\ (a5 Subspaces_of a2);
end;
:: PENCIL_4:def 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being Subspace of b2
for b5 being natural set holds
pencil(b3,b4,b5) = (pencil(b3,b4)) /\ (b5 Subspaces_of b2);
:: PENCIL_4:th 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
for b4, b5, b6 being Subspace of b2
st b6 in pencil(b4,b5,b3)
holds b4 is Subspace of b6 & b6 is Subspace of b5;
:: PENCIL_4:th 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
for b4, b5, b6, b7 being Subspace of b2
st b4 is Subspace of b5 & b6 is Subspace of b7 & (Omega). b4 = (Omega). b6 & (Omega). b5 = (Omega). b7
holds pencil(b4,b5,b3) = pencil(b6,b7,b3);
:: PENCIL_4:funcnot 4 => PENCIL_4:func 4
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3 be natural set;
func A3 Pencils_of A2 -> Element of bool bool (a3 Subspaces_of a2) means
for b1 being set holds
b1 in it
iff
ex b2, b3 being Subspace of a2 st
b2 is Subspace of b3 & (dim b2) + 1 = a3 & dim b3 = a3 + 1 & b1 = pencil(b2,b3,a3);
end;
:: PENCIL_4:def 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
for b4 being Element of bool bool (b3 Subspaces_of b2) holds
b4 = b3 Pencils_of b2
iff
for b5 being set holds
b5 in b4
iff
ex b6, b7 being Subspace of b2 st
b6 is Subspace of b7 & (dim b6) + 1 = b3 & dim b7 = b3 + 1 & b5 = pencil(b6,b7,b3);
:: PENCIL_4:th 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 < dim b2
holds b3 Pencils_of b2 is not empty;
:: PENCIL_4:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4, b5, b6 being Subspace of b2
for b7 being natural set
st 1 <= b7 & b7 < dim b2 & (dim b3) + 1 = b7 & dim b4 = b7 + 1 & b5 in pencil(b3,b4,b7) & b6 in pencil(b3,b4,b7) & b5 <> b6
holds b5 /\ b6 = (Omega). b3 & b5 + b6 = (Omega). b4;
:: PENCIL_4:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being Element of the carrier of b2
st b3 <> 0. b2
holds dim Lin {b3} = 1;
:: PENCIL_4:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b2
st not b4 in b3
holds dim (b3 + Lin {b4}) = (dim b3) + 1;
:: PENCIL_4:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being Subspace of b2
for b4, b5 being Element of the carrier of b2
st not b4 in b3 &
not b5 in b3 &
b4 <> b5 &
{b4,b5} is linearly-independent(b1, b2) &
b3 /\ Lin {b4,b5} = (0). b2
holds dim (b3 + Lin {b4,b5}) = (dim b3) + 2;
:: PENCIL_4:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being Subspace of b2
st b3 is Subspace of b4
for b5 being natural set
st 1 <= b5 & b5 < dim b2 & (dim b3) + 1 = b5 & dim b4 = b5 + 1
for b6 being Element of the carrier of b2
st b6 in b4 & not b6 in b3
holds b3 + Lin {b6} in pencil(b3,b4,b5);
:: PENCIL_4:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being Subspace of b2
st b3 is Subspace of b4
for b5 being natural set
st 1 <= b5 & b5 < dim b2 & (dim b3) + 1 = b5 & dim b4 = b5 + 1
holds pencil(b3,b4,b5) is not trivial;
:: PENCIL_4:funcnot 5 => PENCIL_4:func 5
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3 be natural set;
func PencilSpace(A2,A3) -> strict TopStruct equals
TopStruct(#a3 Subspaces_of a2,a3 Pencils_of a2#);
end;
:: PENCIL_4:def 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set holds
PencilSpace(b2,b3) = TopStruct(#b3 Subspaces_of b2,b3 Pencils_of b2#);
:: PENCIL_4:th 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st b3 <= dim b2
holds PencilSpace(b2,b3) is not empty;
:: PENCIL_4:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 < dim b2
holds PencilSpace(b2,b3) is not void;
:: PENCIL_4:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 < dim b2 & 3 <= dim b2
holds PencilSpace(b2,b3) is not degenerated;
:: PENCIL_4:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 < dim b2
holds PencilSpace(b2,b3) is with_non_trivial_blocks;
:: PENCIL_4:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 < dim b2
holds PencilSpace(b2,b3) is identifying_close_blocks;
:: PENCIL_4:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 < dim b2 & 3 <= dim b2
holds PencilSpace(b2,b3) is non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
:: PENCIL_4:funcnot 6 => PENCIL_4:func 6
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3, a4 be natural set;
func SubspaceSet(A2,A3,A4) -> Element of bool bool (a3 Subspaces_of a2) means
for b1 being set holds
b1 in it
iff
ex b2 being Subspace of a2 st
dim b2 = a4 & b1 = a3 Subspaces_of b2;
end;
:: PENCIL_4:def 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being natural set
for b5 being Element of bool bool (b3 Subspaces_of b2) holds
b5 = SubspaceSet(b2,b3,b4)
iff
for b6 being set holds
b6 in b5
iff
ex b7 being Subspace of b2 st
dim b7 = b4 & b6 = b3 Subspaces_of b7;
:: PENCIL_4:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being natural set
st b4 <= dim b2
holds SubspaceSet(b2,b3,b4) is not empty;
:: PENCIL_4:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
st (Omega). b2 = (Omega). b3
for b4 being natural set holds
b4 Subspaces_of b2 = b4 Subspaces_of b3;
:: PENCIL_4:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being Subspace of b2
for b4 being natural set
st 1 <= b4 & b4 <= dim b2 & b4 Subspaces_of b2 c= b4 Subspaces_of b3
holds (Omega). b2 = (Omega). b3;
:: PENCIL_4:funcnot 7 => PENCIL_4:func 7
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over a1;
let a3, a4 be natural set;
func GrassmannSpace(A2,A3,A4) -> strict TopStruct equals
TopStruct(#a3 Subspaces_of a2,SubspaceSet(a2,a3,a4)#);
end;
:: PENCIL_4:def 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being natural set holds
GrassmannSpace(b2,b3,b4) = TopStruct(#b3 Subspaces_of b2,SubspaceSet(b2,b3,b4)#);
:: PENCIL_4:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being natural set
st b3 <= dim b2
holds GrassmannSpace(b2,b3,b4) is not empty;
:: PENCIL_4:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being natural set
st b4 <= dim b2
holds GrassmannSpace(b2,b3,b4) is not void;
:: PENCIL_4:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being natural set
st 1 <= b3 & b3 < b4 & b4 < dim b2
holds GrassmannSpace(b2,b3,b4) is not degenerated;
:: PENCIL_4:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3, b4 being natural set
st 1 <= b3 & b3 < b4 & b4 <= dim b2
holds GrassmannSpace(b2,b3,b4) is with_non_trivial_blocks;
:: PENCIL_4:th 31
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 + 1 <= dim b2
holds GrassmannSpace(b2,b3,b3 + 1) is identifying_close_blocks;
:: PENCIL_4:th 32
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed finite-dimensional VectSpStr over b1
for b3 being natural set
st 1 <= b3 & b3 + 1 < dim b2
holds GrassmannSpace(b2,b3,b3 + 1) is non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
:: PENCIL_4:funcnot 8 => PENCIL_4:func 8
definition
let a1 be set;
func PairSet A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2, b3 being set st
b2 in a1 & b3 in a1 & b1 = {b2,b3};
end;
:: PENCIL_4:def 8
theorem
for b1, b2 being set holds
b2 = PairSet b1
iff
for b3 being set holds
b3 in b2
iff
ex b4, b5 being set st
b4 in b1 & b5 in b1 & b3 = {b4,b5};
:: PENCIL_4:funcreg 1
registration
let a1 be non empty set;
cluster PairSet a1 -> non empty;
end;
:: PENCIL_4:funcnot 9 => PENCIL_4:func 9
definition
let a1, a2 be set;
func PairSet(A1,A2) -> set means
for b1 being set holds
b1 in it
iff
ex b2 being set st
b2 in a2 & b1 = {a1,b2};
end;
:: PENCIL_4:def 9
theorem
for b1, b2, b3 being set holds
b3 = PairSet(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
ex b5 being set st
b5 in b2 & b4 = {b1,b5};
:: PENCIL_4:funcreg 2
registration
let a1 be set;
let a2 be non empty set;
cluster PairSet(a1,a2) -> non empty;
end;
:: PENCIL_4:funcreg 3
registration
let a1 be set;
let a2 be non trivial set;
cluster PairSet(a1,a2) -> non trivial;
end;
:: PENCIL_4:funcnot 10 => PENCIL_4:func 10
definition
let a1 be set;
let a2 be Element of bool bool a1;
func PairSetFamily A2 -> Element of bool bool PairSet a1 means
for b1 being set holds
b1 in it
iff
ex b2 being set st
ex b3 being Element of bool a1 st
b2 in a1 & b3 in a2 & b1 = PairSet(b2,b3);
end;
:: PENCIL_4:def 10
theorem
for b1 being set
for b2 being Element of bool bool b1
for b3 being Element of bool bool PairSet b1 holds
b3 = PairSetFamily b2
iff
for b4 being set holds
b4 in b3
iff
ex b5 being set st
ex b6 being Element of bool b1 st
b5 in b1 & b6 in b2 & b4 = PairSet(b5,b6);
:: PENCIL_4:funcreg 4
registration
let a1 be non empty set;
let a2 be non empty Element of bool bool a1;
cluster PairSetFamily a2 -> non empty;
end;
:: PENCIL_4:funcnot 11 => PENCIL_4:func 11
definition
let a1 be TopStruct;
func VeroneseSpace A1 -> strict TopStruct equals
TopStruct(#PairSet the carrier of a1,PairSetFamily the topology of a1#);
end;
:: PENCIL_4:def 11
theorem
for b1 being TopStruct holds
VeroneseSpace b1 = TopStruct(#PairSet the carrier of b1,PairSetFamily the topology of b1#);
:: PENCIL_4:funcreg 5
registration
let a1 be non empty TopStruct;
cluster VeroneseSpace a1 -> non empty strict;
end;
:: PENCIL_4:funcreg 6
registration
let a1 be non empty non void TopStruct;
cluster VeroneseSpace a1 -> strict non void;
end;
:: PENCIL_4:funcreg 7
registration
let a1 be non empty non void non degenerated TopStruct;
cluster VeroneseSpace a1 -> strict non degenerated;
end;
:: PENCIL_4:funcreg 8
registration
let a1 be non empty non void with_non_trivial_blocks TopStruct;
cluster VeroneseSpace a1 -> strict with_non_trivial_blocks;
end;
:: PENCIL_4:funcreg 9
registration
let a1 be identifying_close_blocks TopStruct;
cluster VeroneseSpace a1 -> strict identifying_close_blocks;
end;
:: PENCIL_4:funcnot 12 => PENCIL_4:func 12
definition
let a1 be non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
redefine func VeroneseSpace a1 -> non empty strict non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
end;