Article PENCIL_1, MML version 4.99.1005
:: PENCIL_1:th 1
theorem
for b1, b2 being Relation-like Function-like set
st product b1 = product b2 & b1 is non-empty
holds b2 is non-empty;
:: PENCIL_1:th 2
theorem
for b1 being set holds
2 c= Card b1
iff
ex b2, b3 being set st
b2 in b1 & b3 in b1 & b2 <> b3;
:: PENCIL_1:th 3
theorem
for b1 being set
st 2 c= Card b1
for b2 being set holds
ex b3 being set st
b3 in b1 & b2 <> b3;
:: PENCIL_1:th 4
theorem
for b1 being set holds
2 c= Card b1
iff
b1 is not trivial;
:: PENCIL_1:th 5
theorem
for b1 being set holds
3 c= Card b1
iff
ex b2, b3, b4 being set st
b2 in b1 & b3 in b1 & b4 in b1 & b2 <> b3 & b2 <> b4 & b3 <> b4;
:: PENCIL_1:th 6
theorem
for b1 being set
st 3 c= Card b1
for b2, b3 being set holds
ex b4 being set st
b4 in b1 & b2 <> b4 & b3 <> b4;
:: PENCIL_1:modenot 1
definition
let a1 be TopStruct;
mode Block of a1 is Element of the topology of a1;
end;
:: PENCIL_1:prednot 1 => PENCIL_1:pred 1
definition
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
pred A2,A3 are_collinear means
(a2 <> a3) implies ex b1 being Element of the topology of a1 st
{a2,a3} c= b1;
end;
:: PENCIL_1:dfs 1
definiens
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
To prove
a2,a3 are_collinear
it is sufficient to prove
thus (a2 <> a3) implies ex b1 being Element of the topology of a1 st
{a2,a3} c= b1;
:: PENCIL_1:def 1
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_collinear
iff
(b2 <> b3 implies ex b4 being Element of the topology of b1 st
{b2,b3} c= b4);
:: PENCIL_1:attrnot 1 => PENCIL_1:attr 1
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is closed_under_lines means
for b1 being Element of the topology of a1
st 2 c= Card (b1 /\ a2)
holds b1 c= a2;
end;
:: PENCIL_1:dfs 2
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is closed_under_lines
it is sufficient to prove
thus for b1 being Element of the topology of a1
st 2 c= Card (b1 /\ a2)
holds b1 c= a2;
:: PENCIL_1:def 2
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is closed_under_lines(b1)
iff
for b3 being Element of the topology of b1
st 2 c= Card (b3 /\ b2)
holds b3 c= b2;
:: PENCIL_1:attrnot 2 => PENCIL_1:attr 2
definition
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is strong means
for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds b1,b2 are_collinear;
end;
:: PENCIL_1:dfs 3
definiens
let a1 be TopStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is strong
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds b1,b2 are_collinear;
:: PENCIL_1:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
b2 is strong(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b3,b4 are_collinear;
:: PENCIL_1:attrnot 3 => PENCIL_1:attr 3
definition
let a1 be TopStruct;
attr a1 is void means
the topology of a1 is empty;
end;
:: PENCIL_1:dfs 4
definiens
let a1 be TopStruct;
To prove
a1 is void
it is sufficient to prove
thus the topology of a1 is empty;
:: PENCIL_1:def 4
theorem
for b1 being TopStruct holds
b1 is void
iff
the topology of b1 is empty;
:: PENCIL_1:attrnot 4 => PENCIL_1:attr 4
definition
let a1 be TopStruct;
attr a1 is degenerated means
the carrier of a1 is Element of the topology of a1;
end;
:: PENCIL_1:dfs 5
definiens
let a1 be TopStruct;
To prove
a1 is degenerated
it is sufficient to prove
thus the carrier of a1 is Element of the topology of a1;
:: PENCIL_1:def 5
theorem
for b1 being TopStruct holds
b1 is degenerated
iff
the carrier of b1 is Element of the topology of b1;
:: PENCIL_1:attrnot 5 => PENCIL_1:attr 5
definition
let a1 be TopStruct;
attr a1 is with_non_trivial_blocks means
for b1 being Element of the topology of a1 holds
2 c= Card b1;
end;
:: PENCIL_1:dfs 6
definiens
let a1 be TopStruct;
To prove
a1 is with_non_trivial_blocks
it is sufficient to prove
thus for b1 being Element of the topology of a1 holds
2 c= Card b1;
:: PENCIL_1:def 6
theorem
for b1 being TopStruct holds
b1 is with_non_trivial_blocks
iff
for b2 being Element of the topology of b1 holds
2 c= Card b2;
:: PENCIL_1:attrnot 6 => PENCIL_1:attr 6
definition
let a1 be TopStruct;
attr a1 is identifying_close_blocks means
for b1, b2 being Element of the topology of a1
st 2 c= Card (b1 /\ b2)
holds b1 = b2;
end;
:: PENCIL_1:dfs 7
definiens
let a1 be TopStruct;
To prove
a1 is identifying_close_blocks
it is sufficient to prove
thus for b1, b2 being Element of the topology of a1
st 2 c= Card (b1 /\ b2)
holds b1 = b2;
:: PENCIL_1:def 7
theorem
for b1 being TopStruct holds
b1 is identifying_close_blocks
iff
for b2, b3 being Element of the topology of b1
st 2 c= Card (b2 /\ b3)
holds b2 = b3;
:: PENCIL_1:attrnot 7 => PENCIL_1:attr 7
definition
let a1 be TopStruct;
attr a1 is truly-partial means
ex b1, b2 being Element of the carrier of a1 st
not b1,b2 are_collinear;
end;
:: PENCIL_1:dfs 8
definiens
let a1 be TopStruct;
To prove
a1 is truly-partial
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
not b1,b2 are_collinear;
:: PENCIL_1:def 8
theorem
for b1 being TopStruct holds
b1 is truly-partial
iff
ex b2, b3 being Element of the carrier of b1 st
not b2,b3 are_collinear;
:: PENCIL_1:attrnot 8 => PENCIL_1:attr 8
definition
let a1 be TopStruct;
attr a1 is without_isolated_points means
for b1 being Element of the carrier of a1 holds
ex b2 being Element of the topology of a1 st
b1 in b2;
end;
:: PENCIL_1:dfs 9
definiens
let a1 be TopStruct;
To prove
a1 is without_isolated_points
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Element of the topology of a1 st
b1 in b2;
:: PENCIL_1:def 9
theorem
for b1 being TopStruct holds
b1 is without_isolated_points
iff
for b2 being Element of the carrier of b1 holds
ex b3 being Element of the topology of b1 st
b2 in b3;
:: PENCIL_1:attrnot 9 => PENCIL_1:attr 9
definition
let a1 be TopStruct;
attr a1 is connected means
for b1, b2 being Element of the carrier of a1 holds
ex b3 being FinSequence of the carrier of a1 st
b1 = b3 . 1 &
b2 = b3 . len b3 &
(for b4 being natural set
st 1 <= b4 & b4 < len b3
for b5, b6 being Element of the carrier of a1
st b5 = b3 . b4 & b6 = b3 . (b4 + 1)
holds b5,b6 are_collinear);
end;
:: PENCIL_1:dfs 10
definiens
let a1 be TopStruct;
To prove
a1 is connected
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
ex b3 being FinSequence of the carrier of a1 st
b1 = b3 . 1 &
b2 = b3 . len b3 &
(for b4 being natural set
st 1 <= b4 & b4 < len b3
for b5, b6 being Element of the carrier of a1
st b5 = b3 . b4 & b6 = b3 . (b4 + 1)
holds b5,b6 are_collinear);
:: PENCIL_1:def 10
theorem
for b1 being TopStruct holds
b1 is connected
iff
for b2, b3 being Element of the carrier of b1 holds
ex b4 being FinSequence of the carrier of b1 st
b2 = b4 . 1 &
b3 = b4 . len b4 &
(for b5 being natural set
st 1 <= b5 & b5 < len b4
for b6, b7 being Element of the carrier of b1
st b6 = b4 . b5 & b7 = b4 . (b5 + 1)
holds b6,b7 are_collinear);
:: PENCIL_1:attrnot 10 => PENCIL_1:attr 10
definition
let a1 be TopStruct;
attr a1 is strongly_connected means
for b1 being Element of the carrier of a1
for b2 being Element of bool the carrier of a1
st b2 is closed_under_lines(a1) & b2 is strong(a1)
holds ex b3 being FinSequence of bool the carrier of a1 st
b2 = b3 . 1 &
b1 in b3 . len b3 &
(for b4 being Element of bool the carrier of a1
st b4 in proj2 b3
holds b4 is closed_under_lines(a1) & b4 is strong(a1)) &
(for b4 being natural set
st 1 <= b4 & b4 < len b3
holds 2 c= Card ((b3 . b4) /\ (b3 . (b4 + 1))));
end;
:: PENCIL_1:dfs 11
definiens
let a1 be TopStruct;
To prove
a1 is strongly_connected
it is sufficient to prove
thus for b1 being Element of the carrier of a1
for b2 being Element of bool the carrier of a1
st b2 is closed_under_lines(a1) & b2 is strong(a1)
holds ex b3 being FinSequence of bool the carrier of a1 st
b2 = b3 . 1 &
b1 in b3 . len b3 &
(for b4 being Element of bool the carrier of a1
st b4 in proj2 b3
holds b4 is closed_under_lines(a1) & b4 is strong(a1)) &
(for b4 being natural set
st 1 <= b4 & b4 < len b3
holds 2 c= Card ((b3 . b4) /\ (b3 . (b4 + 1))));
:: PENCIL_1:def 11
theorem
for b1 being TopStruct holds
b1 is strongly_connected
iff
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is closed_under_lines(b1) & b3 is strong(b1)
holds ex b4 being FinSequence of bool the carrier of b1 st
b3 = b4 . 1 &
b2 in b4 . len b4 &
(for b5 being Element of bool the carrier of b1
st b5 in proj2 b4
holds b5 is closed_under_lines(b1) & b5 is strong(b1)) &
(for b5 being natural set
st 1 <= b5 & b5 < len b4
holds 2 c= Card ((b4 . b5) /\ (b4 . (b5 + 1))));
:: PENCIL_1:th 7
theorem
for b1 being non empty set
st 3 c= Card b1
for b2 being TopStruct
st the carrier of b2 = b1 &
the topology of b2 = {b3 where b3 is Element of bool b1: 2 = Card b3}
holds b2 is not empty & b2 is not void & b2 is not degenerated & b2 is not truly-partial & b2 is with_non_trivial_blocks & b2 is identifying_close_blocks & b2 is without_isolated_points;
:: PENCIL_1:th 8
theorem
for b1 being non empty set
st 3 c= Card b1
for b2 being Element of bool b1
st Card b2 = 2
for b3 being TopStruct
st the carrier of b3 = b1 &
the topology of b3 = {b4 where b4 is Element of bool b1: 2 = Card b4} \ {b2}
holds b3 is not empty & b3 is not void & b3 is not degenerated & b3 is truly-partial & b3 is with_non_trivial_blocks & b3 is identifying_close_blocks & b3 is without_isolated_points;
:: PENCIL_1:exreg 1
registration
cluster non empty strict non void non degenerated with_non_trivial_blocks identifying_close_blocks non truly-partial without_isolated_points TopStruct;
end;
:: PENCIL_1:exreg 2
registration
cluster non empty strict non void non degenerated with_non_trivial_blocks identifying_close_blocks truly-partial without_isolated_points TopStruct;
end;
:: PENCIL_1:funcreg 1
registration
let a1 be non void TopStruct;
cluster the topology of a1 -> non empty;
end;
:: PENCIL_1:prednot 2 => PENCIL_1:pred 1
definition
let a1 be TopStruct;
let a2, a3 be Element of the carrier of a1;
pred A2,A3 are_collinear means
ex b1 being Element of the topology of a1 st
{a2,a3} c= b1;
end;
:: PENCIL_1:dfs 12
definiens
let a1 be without_isolated_points TopStruct;
let a2, a3 be Element of the carrier of a1;
To prove
a2,a3 are_collinear
it is sufficient to prove
thus ex b1 being Element of the topology of a1 st
{a2,a3} c= b1;
:: PENCIL_1:def 12
theorem
for b1 being without_isolated_points TopStruct
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_collinear
iff
ex b4 being Element of the topology of b1 st
{b2,b3} c= b4;
:: PENCIL_1:modenot 2
definition
mode PLS is non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
end;
:: PENCIL_1:attrnot 11 => PENCIL_1:attr 11
definition
let a1 be Relation-like set;
attr a1 is TopStruct-yielding means
for b1 being set
st b1 in proj2 a1
holds b1 is TopStruct;
end;
:: PENCIL_1:dfs 13
definiens
let a1 be Relation-like set;
To prove
a1 is TopStruct-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj2 a1
holds b1 is TopStruct;
:: PENCIL_1:def 13
theorem
for b1 being Relation-like set holds
b1 is TopStruct-yielding
iff
for b2 being set
st b2 in proj2 b1
holds b2 is TopStruct;
:: PENCIL_1:condreg 1
registration
cluster Relation-like Function-like TopStruct-yielding -> 1-sorted-yielding (set);
end;
:: PENCIL_1:exreg 3
registration
let a1 be set;
cluster Relation-like Function-like TopStruct-yielding ManySortedSet of a1;
end;
:: PENCIL_1:exreg 4
registration
cluster Relation-like Function-like TopStruct-yielding set;
end;
:: PENCIL_1:attrnot 12 => PENCIL_1:attr 12
definition
let a1 be Relation-like set;
attr a1 is non-void-yielding means
for b1 being TopStruct
st b1 in proj2 a1
holds b1 is not void;
end;
:: PENCIL_1:dfs 14
definiens
let a1 be Relation-like set;
To prove
a1 is non-void-yielding
it is sufficient to prove
thus for b1 being TopStruct
st b1 in proj2 a1
holds b1 is not void;
:: PENCIL_1:def 14
theorem
for b1 being Relation-like set holds
b1 is non-void-yielding
iff
for b2 being TopStruct
st b2 in proj2 b1
holds b2 is not void;
:: PENCIL_1:attrnot 13 => PENCIL_1:attr 12
definition
let a1 be Relation-like set;
attr a1 is non-void-yielding means
for b1 being set
st b1 in proj2 a1
holds b1 is non void TopStruct;
end;
:: PENCIL_1:dfs 15
definiens
let a1 be Relation-like Function-like TopStruct-yielding set;
To prove
a1 is non-void-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj2 a1
holds b1 is non void TopStruct;
:: PENCIL_1:def 15
theorem
for b1 being Relation-like Function-like TopStruct-yielding set holds
b1 is non-void-yielding
iff
for b2 being set
st b2 in proj2 b1
holds b2 is non void TopStruct;
:: PENCIL_1:attrnot 14 => PENCIL_1:attr 13
definition
let a1 be Relation-like set;
attr a1 is trivial-yielding means
for b1 being set
st b1 in proj2 a1
holds b1 is trivial;
end;
:: PENCIL_1:dfs 16
definiens
let a1 be Relation-like set;
To prove
a1 is trivial-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj2 a1
holds b1 is trivial;
:: PENCIL_1:def 16
theorem
for b1 being Relation-like set holds
b1 is trivial-yielding
iff
for b2 being set
st b2 in proj2 b1
holds b2 is trivial;
:: PENCIL_1:attrnot 15 => PENCIL_1:attr 14
definition
let a1 be Relation-like set;
attr a1 is non-Trivial-yielding means
for b1 being 1-sorted
st b1 in proj2 a1
holds b1 is not trivial;
end;
:: PENCIL_1:dfs 17
definiens
let a1 be Relation-like set;
To prove
a1 is non-Trivial-yielding
it is sufficient to prove
thus for b1 being 1-sorted
st b1 in proj2 a1
holds b1 is not trivial;
:: PENCIL_1:def 17
theorem
for b1 being Relation-like set holds
b1 is non-Trivial-yielding
iff
for b2 being 1-sorted
st b2 in proj2 b1
holds b2 is not trivial;
:: PENCIL_1:condreg 2
registration
cluster Relation-like non-Trivial-yielding -> non-Empty (set);
end;
:: PENCIL_1:attrnot 16 => PENCIL_1:attr 14
definition
let a1 be Relation-like set;
attr a1 is non-Trivial-yielding means
for b1 being set
st b1 in proj2 a1
holds b1 is non trivial 1-sorted;
end;
:: PENCIL_1:dfs 18
definiens
let a1 be Relation-like Function-like 1-sorted-yielding set;
To prove
a1 is non-Trivial-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj2 a1
holds b1 is non trivial 1-sorted;
:: PENCIL_1:def 18
theorem
for b1 being Relation-like Function-like 1-sorted-yielding set holds
b1 is non-Trivial-yielding
iff
for b2 being set
st b2 in proj2 b1
holds b2 is non trivial 1-sorted;
:: PENCIL_1:funcnot 1 => PENCIL_1:func 1
definition
let a1 be non empty set;
let a2 be TopStruct-yielding ManySortedSet of a1;
let a3 be Element of a1;
redefine func a2 . a3 -> TopStruct;
end;
:: PENCIL_1:attrnot 17 => PENCIL_1:attr 15
definition
let a1 be Relation-like set;
attr a1 is PLS-yielding means
for b1 being set
st b1 in proj2 a1
holds b1 is non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
end;
:: PENCIL_1:dfs 19
definiens
let a1 be Relation-like set;
To prove
a1 is PLS-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj2 a1
holds b1 is non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
:: PENCIL_1:def 19
theorem
for b1 being Relation-like set holds
b1 is PLS-yielding
iff
for b2 being set
st b2 in proj2 b1
holds b2 is non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
:: PENCIL_1:condreg 3
registration
cluster Relation-like Function-like PLS-yielding -> non-Empty TopStruct-yielding (set);
end;
:: PENCIL_1:condreg 4
registration
cluster Relation-like Function-like TopStruct-yielding PLS-yielding -> non-void-yielding (set);
end;
:: PENCIL_1:condreg 5
registration
cluster Relation-like Function-like TopStruct-yielding PLS-yielding -> non-Trivial-yielding (set);
end;
:: PENCIL_1:exreg 5
registration
let a1 be set;
cluster Relation-like Function-like PLS-yielding ManySortedSet of a1;
end;
:: PENCIL_1:funcnot 2 => PENCIL_1:func 2
definition
let a1 be non empty set;
let a2 be PLS-yielding ManySortedSet of a1;
let a3 be Element of a1;
redefine func a2 . a3 -> non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
end;
:: PENCIL_1:attrnot 18 => PENCIL_1:attr 16
definition
let a1 be set;
let a2 be ManySortedSet of a1;
attr a2 is Segre-like means
ex b1 being Element of a1 st
for b2 being Element of a1
st b1 <> b2
holds a2 . b2 is not empty & a2 . b2 is trivial;
end;
:: PENCIL_1:dfs 20
definiens
let a1 be set;
let a2 be ManySortedSet of a1;
To prove
a2 is Segre-like
it is sufficient to prove
thus ex b1 being Element of a1 st
for b2 being Element of a1
st b1 <> b2
holds a2 . b2 is not empty & a2 . b2 is trivial;
:: PENCIL_1:def 20
theorem
for b1 being set
for b2 being ManySortedSet of b1 holds
b2 is Segre-like(b1)
iff
ex b3 being Element of b1 st
for b4 being Element of b1
st b3 <> b4
holds b2 . b4 is not empty & b2 . b4 is trivial;
:: PENCIL_1:funcreg 2
registration
let a1 be set;
let a2 be ManySortedSet of a1;
cluster {a2} -> trivial-yielding;
end;
:: PENCIL_1:th 9
theorem
for b1 being non empty set
for b2 being ManySortedSet of b1
for b3 being Element of b1
for b4 being non trivial set holds
b2 +*(b3,b4) is not trivial-yielding;
:: PENCIL_1:funcreg 3
registration
let a1 be non empty set;
let a2 be ManySortedSet of a1;
cluster {a2} -> Segre-like;
end;
:: PENCIL_1:th 10
theorem
for b1 being non empty set
for b2 being ManySortedSet of b1
for b3, b4 being set holds
{b2} +*(b3,b4) is Segre-like(b1);
:: PENCIL_1:th 11
theorem
for b1 being non empty set
for b2 being 1-sorted-yielding non-Empty ManySortedSet of b1
for b3 being Element of Carrier b2 holds
{b3} is ManySortedSubset of Carrier b2;
:: PENCIL_1:exreg 6
registration
let a1 be non empty set;
let a2 be 1-sorted-yielding non-Empty ManySortedSet of a1;
cluster Relation-like non-empty Function-like trivial-yielding Segre-like ManySortedSubset of Carrier a2;
end;
:: PENCIL_1:exreg 7
registration
let a1 be non empty set;
let a2 be 1-sorted-yielding non-Trivial-yielding ManySortedSet of a1;
cluster Relation-like non-empty Function-like non trivial-yielding Segre-like ManySortedSubset of Carrier a2;
end;
:: PENCIL_1:exreg 8
registration
let a1 be non empty set;
cluster Relation-like Function-like non trivial-yielding Segre-like ManySortedSet of a1;
end;
:: PENCIL_1:funcnot 3 => PENCIL_1:func 3
definition
let a1 be non empty set;
let a2 be non trivial-yielding Segre-like ManySortedSet of a1;
func indx A2 -> Element of a1 means
a2 . it is not trivial;
end;
:: PENCIL_1:def 21
theorem
for b1 being non empty set
for b2 being non trivial-yielding Segre-like ManySortedSet of b1
for b3 being Element of b1 holds
b3 = indx b2
iff
b2 . b3 is not trivial;
:: PENCIL_1:th 12
theorem
for b1 being non empty set
for b2 being non trivial-yielding Segre-like ManySortedSet of b1
for b3 being Element of b1
st b3 <> indx b2
holds b2 . b3 is not empty & b2 . b3 is trivial;
:: PENCIL_1:condreg 6
registration
let a1 be non empty set;
cluster non trivial-yielding Segre-like -> non-empty (ManySortedSet of a1);
end;
:: PENCIL_1:th 13
theorem
for b1 being non empty set
for b2 being ManySortedSet of b1 holds
2 c= Card product b2
iff
b2 is non-empty & b2 is not trivial-yielding;
:: PENCIL_1:funcreg 4
registration
let a1 be non empty set;
let a2 be non trivial-yielding Segre-like ManySortedSet of a1;
cluster product a2 -> non trivial;
end;
:: PENCIL_1:funcnot 4 => PENCIL_1:func 4
definition
let a1 be non empty set;
let a2 be non-Empty TopStruct-yielding ManySortedSet of a1;
func Segre_Blocks A2 -> Element of bool bool product Carrier a2 means
for b1 being set holds
b1 in it
iff
ex b2 being Segre-like ManySortedSubset of Carrier a2 st
b1 = product b2 &
(ex b3 being Element of a1 st
b2 . b3 is Element of the topology of a2 . b3);
end;
:: PENCIL_1:def 22
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1
for b3 being Element of bool bool product Carrier b2 holds
b3 = Segre_Blocks b2
iff
for b4 being set holds
b4 in b3
iff
ex b5 being Segre-like ManySortedSubset of Carrier b2 st
b4 = product b5 &
(ex b6 being Element of b1 st
b5 . b6 is Element of the topology of b2 . b6);
:: PENCIL_1:funcnot 5 => PENCIL_1:func 5
definition
let a1 be non empty set;
let a2 be non-Empty TopStruct-yielding ManySortedSet of a1;
func Segre_Product A2 -> non empty TopStruct equals
TopStruct(#product Carrier a2,Segre_Blocks a2#);
end;
:: PENCIL_1:def 23
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1 holds
Segre_Product b2 = TopStruct(#product Carrier b2,Segre_Blocks b2#);
:: PENCIL_1:th 14
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1
for b3 being Element of the carrier of Segre_Product b2 holds
b3 is ManySortedSet of b1;
:: PENCIL_1:th 15
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1
st ex b3 being Element of b1 st
b2 . b3 is not void
holds Segre_Product b2 is not void;
:: PENCIL_1:th 16
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is not degenerated &
(ex b4 being Element of b1 st
b2 . b4 is not void)
holds Segre_Product b2 is not degenerated;
:: PENCIL_1:th 17
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is with_non_trivial_blocks &
(ex b4 being Element of b1 st
b2 . b4 is not void)
holds Segre_Product b2 is with_non_trivial_blocks;
:: PENCIL_1:th 18
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is identifying_close_blocks &
b2 . b3 is with_non_trivial_blocks &
(ex b4 being Element of b1 st
b2 . b4 is not void)
holds Segre_Product b2 is identifying_close_blocks;
:: PENCIL_1:funcnot 6 => PENCIL_1:func 6
definition
let a1 be non empty set;
let a2 be PLS-yielding ManySortedSet of a1;
redefine func Segre_Product a2 -> non empty non void non degenerated with_non_trivial_blocks identifying_close_blocks TopStruct;
end;
:: PENCIL_1:th 19
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is trivial
holds b2 is strong(b1) & b2 is closed_under_lines(b1);
:: PENCIL_1:th 20
theorem
for b1 being identifying_close_blocks TopStruct
for b2 being Element of the topology of b1
for b3 being Element of bool the carrier of b1
st b3 = b2
holds b3 is closed_under_lines(b1);
:: PENCIL_1:th 21
theorem
for b1 being TopStruct
for b2 being Element of the topology of b1
for b3 being Element of bool the carrier of b1
st b3 = b2
holds b3 is strong(b1);
:: PENCIL_1:th 22
theorem
for b1 being non void TopStruct holds
[#] b1 is closed_under_lines(b1);
:: PENCIL_1:th 23
theorem
for b1 being non empty set
for b2 being non trivial-yielding Segre-like ManySortedSet of b1
for b3, b4 being ManySortedSet of b1
st b3 in product b2 & b4 in product b2
for b5 being set
st b5 <> indx b2
holds b3 . b5 = b4 . b5;
:: PENCIL_1:th 24
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3 being set holds
b3 is Element of the topology of Segre_Product b2
iff
ex b4 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2 st
b3 = product b4 &
b4 . indx b4 is Element of the topology of b2 . indx b4;
:: PENCIL_1:th 25
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3 being ManySortedSet of b1
st b3 is Element of the carrier of Segre_Product b2
for b4 being Element of b1
for b5 being Element of the carrier of b2 . b4 holds
b3 +*(b4,b5) is Element of the carrier of Segre_Product b2;
:: PENCIL_1:th 26
theorem
for b1 being non empty set
for b2, b3 being non trivial-yielding Segre-like ManySortedSet of b1
st 2 c= Card ((product b2) /\ product b3)
holds indx b2 = indx b3 &
(for b4 being set
st b4 <> indx b2
holds b2 . b4 = b3 . b4);
:: PENCIL_1:th 27
theorem
for b1 being non empty set
for b2 being non trivial-yielding Segre-like ManySortedSet of b1
for b3 being non trivial set holds
b2 +*(indx b2,b3) is Segre-like(b1) & b2 +*(indx b2,b3) is not trivial-yielding;
:: PENCIL_1:th 28
theorem
for b1 being non empty non void identifying_close_blocks without_isolated_points TopStruct
st b1 is strongly_connected
holds b1 is connected;
:: PENCIL_1:th 29
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3 being Element of bool the carrier of Segre_Product b2 holds
b3 is not trivial & b3 is strong(Segre_Product b2) & b3 is closed_under_lines(Segre_Product b2)
iff
ex b4 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2 st
b3 = product b4 &
(for b5 being Element of bool the carrier of b2 . indx b4
st b5 = b4 . indx b4
holds b5 is strong(b2 . indx b4) & b5 is closed_under_lines(b2 . indx b4));