Article CARD_3, MML version 4.99.1005
:: CARD_3:attrnot 1 => CARD_3:attr 1
definition
let a1 be Relation-like Function-like set;
attr a1 is Cardinal-yielding means
for b1 being set
st b1 in proj1 a1
holds a1 . b1 is cardinal set;
end;
:: CARD_3:dfs 1
definiens
let a1 be Relation-like Function-like set;
To prove
a1 is Cardinal-yielding
it is sufficient to prove
thus for b1 being set
st b1 in proj1 a1
holds a1 . b1 is cardinal set;
:: CARD_3:def 1
theorem
for b1 being Relation-like Function-like set holds
b1 is Cardinal-yielding
iff
for b2 being set
st b2 in proj1 b1
holds b1 . b2 is cardinal set;
:: CARD_3:exreg 1
registration
cluster Relation-like Function-like Cardinal-yielding set;
end;
:: CARD_3:modenot 1
definition
mode Cardinal-Function is Relation-like Function-like Cardinal-yielding set;
end;
:: CARD_3:funcreg 1
registration
let a1 be Relation-like Function-like Cardinal-yielding set;
let a2 be set;
cluster a1 | a2 -> Relation-like Cardinal-yielding;
end;
:: CARD_3:funcreg 2
registration
let a1 be set;
let a2 be cardinal set;
cluster a1 --> a2 -> Cardinal-yielding;
end;
:: CARD_3:th 3
theorem
{} is Relation-like Function-like Cardinal-yielding set;
:: CARD_3:sch 1
scheme CARD_3:sch 1
{F1 -> set,
F2 -> cardinal set}:
ex b1 being Relation-like Function-like Cardinal-yielding set st
proj1 b1 = F1() &
(for b2 being set
st b2 in F1()
holds b1 . b2 = F2(b2))
:: CARD_3:funcnot 1 => CARD_3:func 1
definition
let a1 be Relation-like Function-like set;
func Card A1 -> Relation-like Function-like Cardinal-yielding set means
proj1 it = proj1 a1 &
(for b1 being set
st b1 in proj1 a1
holds it . b1 = Card (a1 . b1));
end;
:: CARD_3:def 2
theorem
for b1 being Relation-like Function-like set
for b2 being Relation-like Function-like Cardinal-yielding set holds
b2 = Card b1
iff
proj1 b2 = proj1 b1 &
(for b3 being set
st b3 in proj1 b1
holds b2 . b3 = Card (b1 . b3));
:: CARD_3:funcnot 2 => CARD_3:func 2
definition
let a1 be Relation-like Function-like set;
func disjoin A1 -> Relation-like Function-like set means
proj1 it = proj1 a1 &
(for b1 being set
st b1 in proj1 a1
holds it . b1 = [:a1 . b1,{b1}:]);
end;
:: CARD_3:def 3
theorem
for b1, b2 being Relation-like Function-like set holds
b2 = disjoin b1
iff
proj1 b2 = proj1 b1 &
(for b3 being set
st b3 in proj1 b1
holds b2 . b3 = [:b1 . b3,{b3}:]);
:: CARD_3:funcnot 3 => CARD_3:func 3
definition
let a1 be Relation-like Function-like set;
func Union A1 -> set equals
union proj2 a1;
end;
:: CARD_3:def 4
theorem
for b1 being Relation-like Function-like set holds
Union b1 = union proj2 b1;
:: CARD_3:funcnot 4 => CARD_3:func 4
definition
let a1 be Relation-like Function-like set;
func product A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Relation-like Function-like set st
b1 = b2 &
proj1 b2 = proj1 a1 &
(for b3 being set
st b3 in proj1 a1
holds b2 . b3 in a1 . b3);
end;
:: CARD_3:def 5
theorem
for b1 being Relation-like Function-like set
for b2 being set holds
b2 = product b1
iff
for b3 being set holds
b3 in b2
iff
ex b4 being Relation-like Function-like set st
b3 = b4 &
proj1 b4 = proj1 b1 &
(for b5 being set
st b5 in proj1 b1
holds b4 . b5 in b1 . b5);
:: CARD_3:th 8
theorem
for b1 being Relation-like Function-like Cardinal-yielding set holds
Card b1 = b1;
:: CARD_3:th 9
theorem
Card {} = {};
:: CARD_3:th 10
theorem
for b1, b2 being set holds
Card (b1 --> b2) = b1 --> Card b2;
:: CARD_3:th 11
theorem
disjoin {} = {};
:: CARD_3:th 12
theorem
for b1, b2 being set holds
disjoin ({b1} --> b2) = {b1} --> [:b2,{b1}:];
:: CARD_3:th 13
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
st b1 in proj1 b3 & b2 in proj1 b3 & b1 <> b2
holds (disjoin b3) . b1 misses (disjoin b3) . b2;
:: CARD_3:th 15
theorem
for b1, b2 being set holds
Union (b1 --> b2) c= b2;
:: CARD_3:th 16
theorem
for b1, b2 being set
st b1 <> {}
holds Union (b1 --> b2) = b2;
:: CARD_3:th 17
theorem
for b1, b2 being set holds
Union ({b1} --> b2) = b2;
:: CARD_3:th 18
theorem
for b1, b2 being Relation-like Function-like set holds
b1 in product b2
iff
proj1 b1 = proj1 b2 &
(for b3 being set
st b3 in proj1 b2
holds b1 . b3 in b2 . b3);
:: CARD_3:th 19
theorem
product {} = {{}};
:: CARD_3:th 20
theorem
for b1, b2 being set holds
Funcs(b1,b2) = product (b1 --> b2);
:: CARD_3:funcnot 5 => CARD_3:func 5
definition
let a1, a2 be set;
func pi(A2,A1) -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Relation-like Function-like set st
b2 in a2 & b1 = b2 . a1;
end;
:: CARD_3:def 6
theorem
for b1, b2, b3 being set holds
b3 = pi(b2,b1)
iff
for b4 being set holds
b4 in b3
iff
ex b5 being Relation-like Function-like set st
b5 in b2 & b4 = b5 . b1;
:: CARD_3:th 22
theorem
for b1 being set
for b2 being Relation-like Function-like set
st b1 in proj1 b2 & product b2 <> {}
holds pi(product b2,b1) = b2 . b1;
:: CARD_3:th 24
theorem
for b1 being set holds
pi({},b1) = {};
:: CARD_3:th 25
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
pi({b2},b1) = {b2 . b1};
:: CARD_3:th 26
theorem
for b1 being set
for b2, b3 being Relation-like Function-like set holds
pi({b2,b3},b1) = {b2 . b1,b3 . b1};
:: CARD_3:th 27
theorem
for b1, b2, b3 being set holds
pi(b1 \/ b2,b3) = (pi(b1,b3)) \/ pi(b2,b3);
:: CARD_3:th 28
theorem
for b1, b2, b3 being set holds
pi(b1 /\ b2,b3) c= (pi(b1,b3)) /\ pi(b2,b3);
:: CARD_3:th 29
theorem
for b1, b2, b3 being set holds
(pi(b1,b2)) \ pi(b3,b2) c= pi(b1 \ b3,b2);
:: CARD_3:th 30
theorem
for b1, b2, b3 being set holds
(pi(b1,b2)) \+\ pi(b3,b2) c= pi(b1 \+\ b3,b2);
:: CARD_3:th 31
theorem
for b1, b2 being set holds
Card pi(b1,b2) c= Card b1;
:: CARD_3:th 32
theorem
for b1 being set
for b2 being Relation-like Function-like set
st b1 in Union disjoin b2
holds ex b3, b4 being set st
b1 = [b3,b4];
:: CARD_3:th 33
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
b1 in Union disjoin b2
iff
b1 `2 in proj1 b2 & b1 `1 in b2 . (b1 `2) & b1 = [b1 `1,b1 `2];
:: CARD_3:th 34
theorem
for b1, b2 being Relation-like Function-like set
st b1 c= b2
holds disjoin b1 c= disjoin b2;
:: CARD_3:th 35
theorem
for b1, b2 being Relation-like Function-like set
st b1 c= b2
holds Union b1 c= Union b2;
:: CARD_3:th 36
theorem
for b1, b2 being set holds
Union disjoin (b1 --> b2) = [:b2,b1:];
:: CARD_3:th 37
theorem
for b1 being Relation-like Function-like set holds
product b1 = {}
iff
{} in proj2 b1;
:: CARD_3:th 38
theorem
for b1, b2 being Relation-like Function-like set
st proj1 b1 = proj1 b2 &
(for b3 being set
st b3 in proj1 b1
holds b1 . b3 c= b2 . b3)
holds product b1 c= product b2;
:: CARD_3:th 39
theorem
for b1 being Relation-like Function-like Cardinal-yielding set
for b2 being set
st b2 in proj1 b1
holds Card (b1 . b2) = b1 . b2;
:: CARD_3:th 40
theorem
for b1 being Relation-like Function-like Cardinal-yielding set
for b2 being set
st b2 in proj1 b1
holds Card ((disjoin b1) . b2) = b1 . b2;
:: CARD_3:funcnot 6 => CARD_3:func 6
definition
let a1 be Relation-like Function-like Cardinal-yielding set;
func Sum A1 -> cardinal set equals
Card Union disjoin a1;
end;
:: CARD_3:def 7
theorem
for b1 being Relation-like Function-like Cardinal-yielding set holds
Sum b1 = Card Union disjoin b1;
:: CARD_3:funcnot 7 => CARD_3:func 7
definition
let a1 be Relation-like Function-like Cardinal-yielding set;
func Product A1 -> cardinal set equals
Card product a1;
end;
:: CARD_3:def 8
theorem
for b1 being Relation-like Function-like Cardinal-yielding set holds
Product b1 = Card product b1;
:: CARD_3:th 43
theorem
for b1, b2 being Relation-like Function-like Cardinal-yielding set
st proj1 b1 = proj1 b2 &
(for b3 being set
st b3 in proj1 b1
holds b1 . b3 c= b2 . b3)
holds Sum b1 c= Sum b2;
:: CARD_3:th 44
theorem
for b1 being Relation-like Function-like Cardinal-yielding set holds
{} in proj2 b1
iff
Product b1 = 0;
:: CARD_3:th 45
theorem
for b1, b2 being Relation-like Function-like Cardinal-yielding set
st proj1 b1 = proj1 b2 &
(for b3 being set
st b3 in proj1 b1
holds b1 . b3 c= b2 . b3)
holds Product b1 c= Product b2;
:: CARD_3:th 46
theorem
for b1, b2 being Relation-like Function-like Cardinal-yielding set
st b1 c= b2
holds Sum b1 c= Sum b2;
:: CARD_3:th 47
theorem
for b1, b2 being Relation-like Function-like Cardinal-yielding set
st b1 c= b2 & not 0 in proj2 b2
holds Product b1 c= Product b2;
:: CARD_3:th 48
theorem
for b1 being cardinal set holds
Sum ({} --> b1) = 0;
:: CARD_3:th 49
theorem
for b1 being cardinal set holds
Product ({} --> b1) = 1;
:: CARD_3:th 50
theorem
for b1 being cardinal set
for b2 being set holds
Sum ({b2} --> b1) = b1;
:: CARD_3:th 51
theorem
for b1 being cardinal set
for b2 being set holds
Product ({b2} --> b1) = b1;
:: CARD_3:th 52
theorem
for b1, b2 being cardinal set holds
Sum (b1 --> b2) = b1 *` b2;
:: CARD_3:th 53
theorem
for b1, b2 being cardinal set holds
Product (b1 --> b2) = exp(b2,b1);
:: CARD_3:th 54
theorem
for b1 being Relation-like Function-like set holds
Card Union b1 c= Sum Card b1;
:: CARD_3:th 55
theorem
for b1 being Relation-like Function-like Cardinal-yielding set holds
Card Union b1 c= Sum b1;
:: CARD_3:th 56
theorem
for b1, b2 being Relation-like Function-like Cardinal-yielding set
st proj1 b1 = proj1 b2 &
(for b3 being set
st b3 in proj1 b1
holds b1 . b3 in b2 . b3)
holds Sum b1 in Product b2;
:: CARD_3:sch 2
scheme CARD_3:sch 2
{F1 -> finite set}:
ex b1 being set st
b1 in F1() &
(for b2 being set
st b2 in F1() & b2 <> b1
holds not (P1[b2, b1]))
provided
F1() <> {}
and
for b1, b2 being set
st P1[b1, b2] & P1[b2, b1]
holds b1 = b2
and
for b1, b2, b3 being set
st P1[b1, b2] & P1[b2, b3]
holds P1[b1, b3];
:: CARD_3:sch 3
scheme CARD_3:sch 3
{F1 -> finite set}:
ex b1 being set st
b1 in F1() &
(for b2 being set
st b2 in F1()
holds P1[b1, b2])
provided
F1() <> {}
and
for b1, b2 being set
st not (P1[b1, b2])
holds P1[b2, b1]
and
for b1, b2, b3 being set
st P1[b1, b2] & P1[b2, b3]
holds P1[b1, b3];
:: CARD_3:sch 4
scheme CARD_3:sch 4
{F1 -> set,
F2 -> set}:
ex b1 being Relation-like Function-like set st
proj1 b1 = F1() &
(for b2 being set
st b2 in F1()
for b3 being set holds
b3 in b1 . b2
iff
b3 in F2(b2) & P1[b2, b3])
:: CARD_3:th 57
theorem
for b1 being Element of NAT holds
Rank b1 is finite;
:: CARD_3:th 58
theorem
for b1 being set
st b1 is finite
holds Card b1 in Card omega;
:: CARD_3:th 59
theorem
for b1, b2 being ordinal set
st Card b1 in Card b2
holds b1 in b2;
:: CARD_3:th 60
theorem
for b1 being ordinal set
for b2 being cardinal set
st Card b1 in b2
holds b1 in b2;
:: CARD_3:th 61
theorem
for b1 being set
st b1 is c=-linear
holds ex b2 being set st
b2 c= b1 &
union b2 = union b1 &
(for b3 being set
st b3 c= b2 & b3 <> {}
holds ex b4 being set st
b4 in b3 &
(for b5 being set
st b5 in b3
holds b4 c= b5));
:: CARD_3:th 62
theorem
for b1 being cardinal set
for b2 being set
st (for b3 being set
st b3 in b2
holds Card b3 in b1) &
b2 is c=-linear
holds Card union b2 c= b1;
:: CARD_3:funcreg 3
registration
let a1 be Relation-like Function-like set;
cluster product a1 -> functional;
end;
:: CARD_3:condreg 1
registration
let a1 be set;
let a2 be with_non-empty_elements set;
cluster Function-like quasi_total -> non-empty (Relation of a1,a2);
end;
:: CARD_3:funcreg 4
registration
let a1 be Relation-like non-empty Function-like set;
cluster product a1 -> non empty;
end;
:: CARD_3:th 63
theorem
for b1, b2, b3, b4 being set
st b1 <> b2
holds product ((b1,b2)-->({b3},{b4})) = {(b1,b2)-->(b3,b4)};
:: CARD_3:th 64
theorem
for b1 being set
for b2 being Relation-like Function-like set
st b1 in product b2
holds b1 is Relation-like Function-like set;
:: CARD_3:funcnot 8 => CARD_3:func 8
definition
let a1 be Relation-like Function-like set;
func sproduct A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Relation-like Function-like set st
b1 = b2 &
proj1 b2 c= proj1 a1 &
(for b3 being set
st b3 in proj1 b2
holds b2 . b3 in a1 . b3);
end;
:: CARD_3:def 9
theorem
for b1 being Relation-like Function-like set
for b2 being set holds
b2 = sproduct b1
iff
for b3 being set holds
b3 in b2
iff
ex b4 being Relation-like Function-like set st
b3 = b4 &
proj1 b4 c= proj1 b1 &
(for b5 being set
st b5 in proj1 b4
holds b4 . b5 in b1 . b5);
:: CARD_3:funcreg 5
registration
let a1 be Relation-like Function-like set;
cluster sproduct a1 -> non empty functional;
end;
:: CARD_3:th 65
theorem
for b1, b2 being Relation-like Function-like set
st b1 in sproduct b2
holds proj1 b1 c= proj1 b2 &
(for b3 being set
st b3 in proj1 b1
holds b1 . b3 in b2 . b3);
:: CARD_3:th 66
theorem
for b1 being Relation-like Function-like set holds
{} in sproduct b1;
:: CARD_3:th 67
theorem
for b1 being Relation-like Function-like set holds
product b1 c= sproduct b1;
:: CARD_3:th 68
theorem
for b1 being set
for b2 being Relation-like Function-like set
st b1 in sproduct b2
holds b1 is Function-like Relation of proj1 b2,union proj2 b2;
:: CARD_3:th 69
theorem
for b1, b2, b3 being Relation-like Function-like set
st b1 in product b2 & b3 in sproduct b2
holds b1 +* b3 in product b2;
:: CARD_3:th 70
theorem
for b1, b2 being Relation-like Function-like set
st product b1 <> {}
holds b2 in sproduct b1
iff
ex b3 being Relation-like Function-like set st
b3 in product b1 & b2 c= b3;
:: CARD_3:th 71
theorem
for b1 being Relation-like Function-like set holds
sproduct b1 c= PFuncs(proj1 b1,union proj2 b1);
:: CARD_3:th 72
theorem
for b1, b2 being Relation-like Function-like set
st b1 c= b2
holds sproduct b1 c= sproduct b2;
:: CARD_3:th 73
theorem
sproduct {} = {{}};
:: CARD_3:th 74
theorem
for b1, b2 being set holds
PFuncs(b1,b2) = sproduct (b1 --> b2);
:: CARD_3:th 75
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,b2 holds
sproduct b3 = sproduct (b3 | {b4 where b4 is Element of b1: b3 . b4 <> {}});
:: CARD_3:th 76
theorem
for b1, b2 being set
for b3 being Relation-like Function-like set
st b1 in proj1 b3 & b2 in b3 . b1
holds b1 .--> b2 in sproduct b3;
:: CARD_3:th 77
theorem
for b1 being Relation-like Function-like set holds
sproduct b1 = {{}}
iff
for b2 being set
st b2 in proj1 b1
holds b1 . b2 = {};
:: CARD_3:th 78
theorem
for b1 being Relation-like Function-like set
for b2 being set
st b2 c= sproduct b1 &
(for b3, b4 being Relation-like Function-like set
st b3 in b2 & b4 in b2
holds b3 tolerates b4)
holds union b2 in sproduct b1;
:: CARD_3:th 79
theorem
for b1, b2, b3 being Relation-like Function-like set
st b1 tolerates b2 & b1 in sproduct b3 & b2 in sproduct b3
holds b1 \/ b2 in sproduct b3;
:: CARD_3:th 80
theorem
for b1, b2, b3 being Relation-like Function-like set
st b1 c= b2 & b2 in sproduct b3
holds b1 in sproduct b3;
:: CARD_3:th 81
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set
st b1 in sproduct b2
holds b1 | b3 in sproduct b2;
:: CARD_3:th 82
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set
st b1 in sproduct b2
holds b1 | b3 in sproduct (b2 | b3);
:: CARD_3:th 83
theorem
for b1, b2, b3 being Relation-like Function-like set
st b1 in sproduct (b2 +* b3)
holds ex b4, b5 being Relation-like Function-like set st
b4 in sproduct b2 & b5 in sproduct b3 & b1 = b4 +* b5;
:: CARD_3:th 84
theorem
for b1, b2, b3, b4 being Relation-like Function-like set
st proj1 b1 misses (proj1 b3) \ proj1 b4 & b3 in sproduct b2 & b4 in sproduct b1
holds b3 +* b4 in sproduct (b2 +* b1);
:: CARD_3:th 85
theorem
for b1, b2, b3, b4 being Relation-like Function-like set
st proj1 b3 misses (proj1 b1) \ proj1 b4 & b3 in sproduct b2 & b4 in sproduct b1
holds b3 +* b4 in sproduct (b2 +* b1);
:: CARD_3:th 86
theorem
for b1, b2, b3 being Relation-like Function-like set
st b1 in sproduct b2 & b3 in sproduct b2
holds b1 +* b3 in sproduct b2;
:: CARD_3:th 87
theorem
for b1 being Relation-like Function-like set
for b2, b3, b4, b5 being set
st b2 in proj1 b1 & b4 in b1 . b2 & b3 in proj1 b1 & b5 in b1 . b3
holds (b2,b3)-->(b4,b5) in sproduct b1;
:: CARD_3:attrnot 2 => CARD_3:attr 2
definition
let a1 be set;
attr a1 is with_common_domain means
for b1, b2 being Relation-like Function-like set
st b1 in a1 & b2 in a1
holds proj1 b1 = proj1 b2;
end;
:: CARD_3:dfs 10
definiens
let a1 be set;
To prove
a1 is with_common_domain
it is sufficient to prove
thus for b1, b2 being Relation-like Function-like set
st b1 in a1 & b2 in a1
holds proj1 b1 = proj1 b2;
:: CARD_3:def 10
theorem
for b1 being set holds
b1 is with_common_domain
iff
for b2, b3 being Relation-like Function-like set
st b2 in b1 & b3 in b1
holds proj1 b2 = proj1 b3;
:: CARD_3:exreg 2
registration
cluster non empty functional with_common_domain set;
end;
:: CARD_3:th 88
theorem
{{}} is functional with_common_domain set;
:: CARD_3:funcnot 9 => CARD_3:func 9
definition
let a1 be functional with_common_domain set;
func DOM A1 -> set means
for b1 being Relation-like Function-like set
st b1 in a1
holds it = proj1 b1
if a1 <> {}
otherwise it = {};
end;
:: CARD_3:def 12
theorem
for b1 being functional with_common_domain set
for b2 being set holds
(b1 = {} or (b2 = DOM b1
iff
for b3 being Relation-like Function-like set
st b3 in b1
holds b2 = proj1 b3)) &
(b1 = {} implies (b2 = DOM b1
iff
b2 = {}));
:: CARD_3:th 89
theorem
for b1 being functional with_common_domain set
st b1 = {{}}
holds DOM b1 = {};
:: CARD_3:funcnot 10 => CARD_3:func 10
definition
let a1 be functional set;
func product" A1 -> Relation-like Function-like set means
(for b1 being set holds
b1 in proj1 it
iff
for b2 being Relation-like Function-like set
st b2 in a1
holds b1 in proj1 b2) &
(for b1 being set
st b1 in proj1 it
holds it . b1 = pi(a1,b1))
if a1 is not empty
otherwise it = {};
end;
:: CARD_3:def 13
theorem
for b1 being functional set
for b2 being Relation-like Function-like set holds
(b1 is empty or (b2 = product" b1
iff
(for b3 being set holds
b3 in proj1 b2
iff
for b4 being Relation-like Function-like set
st b4 in b1
holds b3 in proj1 b4) &
(for b3 being set
st b3 in proj1 b2
holds b2 . b3 = pi(b1,b3)))) &
(b1 is empty implies (b2 = product" b1
iff
b2 = {}));
:: CARD_3:th 90
theorem
for b1 being non empty functional set holds
proj1 product" b1 = meet {proj1 b2 where b2 is Element of b1: TRUE};
:: CARD_3:th 91
theorem
for b1 being non empty functional set
for b2 being set
st b2 in proj1 product" b1
holds (product" b1) . b2 = {b3 . b2 where b3 is Element of b1: TRUE};
:: CARD_3:attrnot 3 => CARD_3:attr 3
definition
let a1 be set;
attr a1 is product-like means
ex b1 being Relation-like Function-like set st
a1 = product b1;
end;
:: CARD_3:dfs 13
definiens
let a1 be set;
To prove
a1 is product-like
it is sufficient to prove
thus ex b1 being Relation-like Function-like set st
a1 = product b1;
:: CARD_3:def 14
theorem
for b1 being set holds
b1 is product-like
iff
ex b2 being Relation-like Function-like set st
b1 = product b2;
:: CARD_3:funcreg 6
registration
let a1 be Relation-like Function-like set;
cluster product a1 -> product-like;
end;
:: CARD_3:condreg 2
registration
cluster product-like -> functional with_common_domain (set);
end;
:: CARD_3:exreg 3
registration
cluster non empty product-like set;
end;
:: CARD_3:th 92
theorem
for b1 being functional with_common_domain set holds
proj1 product" b1 = DOM b1;
:: CARD_3:th 93
theorem
for b1 being functional set
for b2 being set
st b2 in proj1 product" b1
holds (product" b1) . b2 = pi(b1,b2);
:: CARD_3:th 94
theorem
for b1 being functional with_common_domain set holds
b1 c= product product" b1;
:: CARD_3:th 95
theorem
for b1 being non empty product-like set holds
b1 = product product" b1;