Article PENCIL_2, MML version 4.99.1005
:: PENCIL_2:funcnot 1 => PENCIL_2:func 1
definition
let a1 be set;
let a2 be FinSequence of a1;
let a3, a4 be natural set;
func Del(A2,A3,A4) -> FinSequence of a1 equals
(a2 | (a3 -' 1)) ^ (a2 /^ a4);
end;
:: PENCIL_2:def 1
theorem
for b1 being set
for b2 being FinSequence of b1
for b3, b4 being natural set holds
Del(b2,b3,b4) = (b2 | (b3 -' 1)) ^ (b2 /^ b4);
:: PENCIL_2:th 1
theorem
for b1 being set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT holds
rng Del(b2,b3,b4) c= rng b2;
:: PENCIL_2:th 2
theorem
for b1 being set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st b3 in dom b2 & b4 in dom b2
holds len Del(b2,b3,b4) = (((len b2) - b4) + b3) - 1;
:: PENCIL_2:th 3
theorem
for b1 being set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st b3 in dom b2 & b4 in dom b2 & len Del(b2,b3,b4) = {}
holds b3 = 1 & b4 = len b2;
:: PENCIL_2:th 4
theorem
for b1 being set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of NAT
st b3 in dom b2 & 1 <= b5 & b5 <= b3 - 1
holds (Del(b2,b3,b4)) . b5 = b2 . b5;
:: PENCIL_2:th 5
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being Element of NAT
st (len b1) + 1 <= b3
holds (b1 ^ b2) . b3 = b2 . (b3 - len b1);
:: PENCIL_2:th 6
theorem
for b1 being set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of NAT
st b3 in dom b2 &
b4 in dom b2 &
b3 <= b4 &
b3 <= b5 &
b5 <= (((len b2) - b4) + b3) - 1
holds (Del(b2,b3,b4)) . b5 = b2 . (((b4 -' b3) + b5) + 1);
:: PENCIL_2:sch 1
scheme PENCIL_2:sch 1
{F1 -> set,
F2 -> set,
F3 -> set,
F4 -> FinSequence of F3()}:
ex b1 being one-to-one FinSequence of F3() st
F1() = b1 . 1 &
F2() = b1 . len b1 &
rng b1 c= rng F4() &
(for b2 being Element of NAT
st 1 <= b2 & b2 < len b1
holds P1[b1 . b2, b1 . (b2 + 1)])
provided
F1() = F4() . 1 & F2() = F4() . len F4()
and
for b1 being Element of NAT
for b2, b3 being set
st 1 <= b1 & b1 < len F4() & b2 = F4() . b1 & b3 = F4() . (b1 + 1)
holds P1[b2, b3];
:: PENCIL_2:th 7
theorem
for b1 being non empty set
for b2 being 1-sorted-yielding ManySortedSet of b1
for b3 being ManySortedSubset of Carrier b2
for b4 being Element of b1
for b5 being Element of bool the carrier of b2 . b4 holds
b3 +*(b4,b5) is ManySortedSubset of Carrier b2;
:: PENCIL_2:modenot 1 => PENCIL_2:mode 1
definition
let a1 be non empty set;
let a2 be TopStruct-yielding non-Trivial-yielding ManySortedSet of a1;
mode Segre-Coset of A2 -> Element of bool the carrier of Segre_Product a2 means
ex b1 being non trivial-yielding Segre-like ManySortedSubset of Carrier a2 st
it = product b1 &
b1 . indx b1 = [#] (a2 . indx b1);
end;
:: PENCIL_2:dfs 2
definiens
let a1 be non empty set;
let a2 be TopStruct-yielding non-Trivial-yielding ManySortedSet of a1;
let a3 be Element of bool the carrier of Segre_Product a2;
To prove
a3 is Segre-Coset of a2
it is sufficient to prove
thus ex b1 being non trivial-yielding Segre-like ManySortedSubset of Carrier a2 st
a3 = product b1 &
b1 . indx b1 = [#] (a2 . indx b1);
:: PENCIL_2:def 2
theorem
for b1 being non empty set
for b2 being TopStruct-yielding non-Trivial-yielding ManySortedSet of b1
for b3 being Element of bool the carrier of Segre_Product b2 holds
b3 is Segre-Coset of b2
iff
ex b4 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2 st
b3 = product b4 &
b4 . indx b4 = [#] (b2 . indx b4);
:: PENCIL_2:th 8
theorem
for b1 being non empty set
for b2 being TopStruct-yielding non-Trivial-yielding ManySortedSet of b1
for b3, b4 being Segre-Coset of b2
st 2 c= Card (b3 /\ b4)
holds b3 = b4;
:: PENCIL_2:prednot 1 => PENCIL_2:pred 1
definition
let a1 be TopStruct;
let a2, a3 be Element of bool the carrier of a1;
pred A2,A3 are_joinable means
ex b1 being FinSequence of bool the carrier of a1 st
a2 = b1 . 1 &
a3 = b1 . len b1 &
(for b2 being Element of bool the carrier of a1
st b2 in rng b1
holds b2 is closed_under_lines(a1) & b2 is strong(a1)) &
(for b2 being Element of NAT
st 1 <= b2 & b2 < len b1
holds 2 c= Card ((b1 . b2) /\ (b1 . (b2 + 1))));
end;
:: PENCIL_2:dfs 3
definiens
let a1 be TopStruct;
let a2, a3 be Element of bool the carrier of a1;
To prove
a2,a3 are_joinable
it is sufficient to prove
thus ex b1 being FinSequence of bool the carrier of a1 st
a2 = b1 . 1 &
a3 = b1 . len b1 &
(for b2 being Element of bool the carrier of a1
st b2 in rng b1
holds b2 is closed_under_lines(a1) & b2 is strong(a1)) &
(for b2 being Element of NAT
st 1 <= b2 & b2 < len b1
holds 2 c= Card ((b1 . b2) /\ (b1 . (b2 + 1))));
:: PENCIL_2:def 3
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2,b3 are_joinable
iff
ex b4 being FinSequence of bool the carrier of b1 st
b2 = b4 . 1 &
b3 = b4 . len b4 &
(for b5 being Element of bool the carrier of b1
st b5 in rng b4
holds b5 is closed_under_lines(b1) & b5 is strong(b1)) &
(for b5 being Element of NAT
st 1 <= b5 & b5 < len b4
holds 2 c= Card ((b4 . b5) /\ (b4 . (b5 + 1))));
:: PENCIL_2:th 9
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
st b2,b3 are_joinable
holds ex b4 being one-to-one FinSequence of bool the carrier of b1 st
b2 = b4 . 1 &
b3 = b4 . len b4 &
(for b5 being Element of bool the carrier of b1
st b5 in rng b4
holds b5 is closed_under_lines(b1) & b5 is strong(b1)) &
(for b5 being Element of NAT
st 1 <= b5 & b5 < len b4
holds 2 c= Card ((b4 . b5) /\ (b4 . (b5 + 1))));
:: PENCIL_2:th 10
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1
st b2 is closed_under_lines(b1) & b2 is strong(b1)
holds b2,b2 are_joinable;
:: PENCIL_2:th 11
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3, b4 being Element of bool the carrier of Segre_Product b2
st b3 is not trivial & b3 is closed_under_lines(Segre_Product b2) & b3 is strong(Segre_Product b2) & b4 is not trivial & b4 is closed_under_lines(Segre_Product b2) & b4 is strong(Segre_Product b2) & b3,b4 are_joinable
for b5, b6 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
st b3 = product b5 & b4 = product b6
holds indx b5 = indx b6 &
(for b7 being set
st b7 <> indx b5
holds b5 . b7 = b6 . b7);
:: PENCIL_2:th 12
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is bijective(the carrier of b1, the carrier of b2)
holds b3 /" is bijective(the carrier of b2, the carrier of b1);
:: PENCIL_2:attrnot 1 => PENCIL_2:attr 1
definition
let a1, a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is isomorphic means
a3 is bijective(the carrier of a1, the carrier of a2) & a3 is open(a1, a2) & a3 /" is bijective(the carrier of a2, the carrier of a1) & a3 /" is open(a2, a1);
end;
:: PENCIL_2:dfs 4
definiens
let a1, a2 be TopStruct;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is isomorphic
it is sufficient to prove
thus a3 is bijective(the carrier of a1, the carrier of a2) & a3 is open(a1, a2) & a3 /" is bijective(the carrier of a2, the carrier of a1) & a3 /" is open(a2, a1);
:: PENCIL_2:def 4
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is isomorphic(b1, b2)
iff
b3 is bijective(the carrier of b1, the carrier of b2) & b3 is open(b1, b2) & b3 /" is bijective(the carrier of b2, the carrier of b1) & b3 /" is open(b2, b1);
:: PENCIL_2:exreg 1
registration
let a1 be non empty TopStruct;
cluster Relation-like Function-like non empty quasi_total total isomorphic Relation of the carrier of a1,the carrier of a1;
end;
:: PENCIL_2:modenot 2
definition
let a1 be non empty TopStruct;
mode Collineation of a1 is Function-like quasi_total isomorphic Relation of the carrier of a1,the carrier of a1;
end;
:: PENCIL_2:funcnot 2 => PENCIL_2:func 2
definition
let a1 be non empty non void TopStruct;
let a2 be Function-like quasi_total isomorphic Relation of the carrier of a1,the carrier of a1;
let a3 be Element of the topology of a1;
redefine func a2 .: a3 -> Element of the topology of a1;
end;
:: PENCIL_2:funcnot 3 => PENCIL_2:func 3
definition
let a1 be non empty non void TopStruct;
let a2 be Function-like quasi_total isomorphic Relation of the carrier of a1,the carrier of a1;
let a3 be Element of the topology of a1;
redefine func a2 " a3 -> Element of the topology of a1;
end;
:: PENCIL_2:th 13
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1 holds
b2 /" is Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1;
:: PENCIL_2:th 14
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is not trivial
holds b2 .: b3 is not trivial;
:: PENCIL_2:th 15
theorem
for b1 being non empty TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is not trivial
holds b2 " b3 is not trivial;
:: PENCIL_2:th 16
theorem
for b1 being non empty non void TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is strong(b1)
holds b2 .: b3 is strong(b1);
:: PENCIL_2:th 17
theorem
for b1 being non empty non void TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is strong(b1)
holds b2 " b3 is strong(b1);
:: PENCIL_2:th 18
theorem
for b1 being non empty non void TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is closed_under_lines(b1)
holds b2 .: b3 is closed_under_lines(b1);
:: PENCIL_2:th 19
theorem
for b1 being non empty non void TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3 being Element of bool the carrier of b1
st b3 is closed_under_lines(b1)
holds b2 " b3 is closed_under_lines(b1);
:: PENCIL_2:th 20
theorem
for b1 being non empty non void TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 is not trivial & b4 is not trivial & b3,b4 are_joinable
holds b2 .: b3,b2 .: b4 are_joinable;
:: PENCIL_2:th 21
theorem
for b1 being non empty non void TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 is not trivial & b4 is not trivial & b3,b4 are_joinable
holds b2 " b3,b2 " b4 are_joinable;
:: PENCIL_2:th 22
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is strongly_connected
for b3 being Element of bool the carrier of Segre_Product b2
st b3 is not trivial & b3 is strong(Segre_Product b2) & b3 is closed_under_lines(Segre_Product b2)
holds union {b4 where b4 is Element of bool the carrier of Segre_Product b2: b4 is not trivial & b4 is strong(Segre_Product b2) & b4 is closed_under_lines(Segre_Product b2) & b3,b4 are_joinable} is Segre-Coset of b2;
:: PENCIL_2:th 23
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is strongly_connected
for b3 being set holds
b3 is Segre-Coset of b2
iff
ex b4 being Element of bool the carrier of Segre_Product b2 st
b4 is not trivial &
b4 is strong(Segre_Product b2) &
b4 is closed_under_lines(Segre_Product b2) &
b3 = union {b5 where b5 is Element of bool the carrier of Segre_Product b2: b5 is not trivial & b5 is strong(Segre_Product b2) & b5 is closed_under_lines(Segre_Product b2) & b4,b5 are_joinable};
:: PENCIL_2:th 24
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is strongly_connected
for b3 being Segre-Coset of b2
for b4 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2 holds
b4 .: b3 is Segre-Coset of b2;
:: PENCIL_2:th 25
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
st for b3 being Element of b1 holds
b2 . b3 is strongly_connected
for b3 being Segre-Coset of b2
for b4 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2 holds
b4 " b3 is Segre-Coset of b2;