begin
theorem Th1:
scheme
FuncIdxcorrectness{
F1()
-> set ,
F2()
-> non
empty set ,
F3(
set )
-> Element of
F2() } :
( ex
F being
Function of
F1(),
F2() st
for
x being
set st
x in F1() holds
F /. x = F3(
x) & ( for
F1,
F2 being
Function of
F1(),
F2() st ( for
x being
set st
x in F1() holds
F1 /. x = F3(
x) ) & ( for
x being
set st
x in F1() holds
F2 /. x = F3(
x) ) holds
F1 = F2 ) )
theorem Th2:
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th7:
for
x1,
x2 being
set for
A being non
empty set st
x1 <> x2 holds
for
y1,
y2 being
Element of
A holds
(
(x1,x2 --> y1,y2) /. x1 = y1 &
(x1,x2 --> y1,y2) /. x2 = y2 )
begin
:: deftheorem CAT_3:def 1 :
canceled;
:: deftheorem CAT_3:def 2 :
canceled;
:: deftheorem Def3 defines doms CAT_3:def 3 :
:: deftheorem Def4 defines cods CAT_3:def 4 :
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
:: deftheorem Def5 defines opp CAT_3:def 5 :
theorem
theorem
theorem
:: deftheorem Def6 defines opp CAT_3:def 6 :
theorem
theorem
theorem
:: deftheorem Def7 defines * CAT_3:def 7 :
:: deftheorem Def8 defines * CAT_3:def 8 :
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
:: deftheorem Def9 defines "*" CAT_3:def 9 :
theorem Th22:
theorem
for
x1,
x2 being
set for
C being
Category for
p1,
p2,
q1,
q2 being
Morphism of
C st
x1 <> x2 holds
(x1,x2 --> p1,p2) "*" (x1,x2 --> q1,q2) = x1,
x2 --> (p1 * q1),
(p2 * q2)
theorem
theorem
begin
:: deftheorem Def10 defines retraction CAT_3:def 10 :
:: deftheorem Def11 defines coretraction CAT_3:def 11 :
theorem Th26:
theorem
theorem
theorem
theorem
theorem
theorem
theorem Th33:
theorem
theorem
theorem
theorem
theorem
begin
:: deftheorem defines term CAT_3:def 12 :
theorem Th39:
theorem Th40:
theorem
begin
:: deftheorem defines init CAT_3:def 13 :
theorem Th42:
theorem Th43:
theorem
begin
:: deftheorem Def14 defines Projections_family CAT_3:def 14 :
theorem Th45:
theorem Th46:
theorem Th47:
theorem Th48:
theorem
canceled;
theorem Th50:
theorem
theorem
:: deftheorem Def15 defines is_a_product_wrt CAT_3:def 15 :
theorem Th53:
theorem Th54:
theorem Th55:
theorem Th56:
theorem
theorem
:: deftheorem Def16 defines is_a_product_wrt CAT_3:def 16 :
theorem Th59:
theorem
for
C being
Category for
c,
a,
b being
Object of
C st
Hom c,
a <> {} &
Hom c,
b <> {} holds
for
p1 being
Morphism of
c,
a for
p2 being
Morphism of
c,
b holds
(
c is_a_product_wrt p1,
p2 iff for
d being
Object of
C st
Hom d,
a <> {} &
Hom d,
b <> {} holds
(
Hom d,
c <> {} & ( for
f being
Morphism of
d,
a for
g being
Morphism of
d,
b ex
h being
Morphism of
d,
c st
for
k being
Morphism of
d,
c holds
( (
p1 * k = f &
p2 * k = g ) iff
h = k ) ) ) )
theorem
theorem
theorem Th63:
theorem
theorem
theorem
begin
:: deftheorem Def17 defines Injections_family CAT_3:def 17 :
theorem Th67:
theorem
theorem Th69:
theorem Th70:
theorem
canceled;
theorem Th72:
theorem
theorem Th74:
theorem Th75:
theorem
:: deftheorem Def18 defines is_a_coproduct_wrt CAT_3:def 18 :
theorem
theorem Th78:
theorem Th79:
theorem Th80:
theorem Th81:
theorem
theorem
:: deftheorem Def19 defines is_a_coproduct_wrt CAT_3:def 19 :
theorem
theorem Th85:
theorem
theorem
for
C being
Category for
a,
c,
b being
Object of
C st
Hom a,
c <> {} &
Hom b,
c <> {} holds
for
i1 being
Morphism of
a,
c for
i2 being
Morphism of
b,
c holds
(
c is_a_coproduct_wrt i1,
i2 iff for
d being
Object of
C st
Hom a,
d <> {} &
Hom b,
d <> {} holds
(
Hom c,
d <> {} & ( for
f being
Morphism of
a,
d for
g being
Morphism of
b,
d ex
h being
Morphism of
c,
d st
for
k being
Morphism of
c,
d holds
( (
k * i1 = f &
k * i2 = g ) iff
h = k ) ) ) )
theorem
theorem Th89:
theorem
theorem
theorem