begin
theorem
for
I being ( ( ) ( )
set )
for
A being ( ( non
empty ) ( non
empty )
set )
for
F1,
F2 being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) ,
A : ( ( non
empty ) ( non
empty )
set ) ) st ( for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) holds
F1 : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of
b2 : ( ( non
empty ) ( non
empty )
set ) )
= F2 : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F1 : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
= F2 : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ;
scheme
FuncIdxcorrectness{
F1()
-> ( ( ) ( )
set ) ,
F2()
-> ( ( non
empty ) ( non
empty )
set ) ,
F3( ( ( ) ( )
set ) )
-> ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) } :
( ex
F being ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) st
for
x being ( ( ) ( )
set ) st
x : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
in F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
F : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
/. x : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
= F3( ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ,
x : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) & ( for
F1,
F2 being ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) st ( for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
F1 : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
= F3( ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ,
x : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) & ( for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) holds
F2 : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
= F3( ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ,
x : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) holds
F1 : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) )
= F2 : ( (
Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
F1( ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
F2( ( ( non
empty ) ( non
empty )
set ) ) : ( ( non
empty ) ( non
empty )
set ) ) ) )
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
A being ( ( non
empty ) ( non
empty )
set ) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
for
y1,
y2 being ( ( ) ( )
Element of
A : ( ( non
empty ) ( non
empty )
set ) ) holds
(
((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (y1 : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) ,y2 : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/. x1 : ( ( ) ( )
set ) : ( ( ) ( )
Element of
b3 : ( ( non
empty ) ( non
empty )
set ) )
= y1 : ( ( ) ( )
Element of
b3 : ( ( non
empty ) ( non
empty )
set ) ) &
((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (y1 : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) ,y2 : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/. x2 : ( ( ) ( )
set ) : ( ( ) ( )
Element of
b3 : ( ( non
empty ) ( non
empty )
set ) )
= y2 : ( ( ) ( )
Element of
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) ;
begin
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) holds
doms ((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
(dom p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(dom p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) holds
cods ((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
(cod p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(cod p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
opp : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
= (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
(p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) opp) : ( ( ) ( )
Element of the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ,
(p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) opp) : ( ( ) ( )
Element of the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
(F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) opp) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
opp : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
((b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non empty non void strict ) ( non empty non void V52() strict Category-like V65() V66() V67() with_identities ) CatStr ) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
((b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non empty non void strict ) ( non empty non void V52() strict Category-like V65() V66() V67() with_identities ) CatStr ) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
((b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non empty non void strict ) ( non empty non void V52() strict Category-like V65() V66() V67() with_identities ) CatStr ) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
((b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non empty non void strict ) ( non empty non void V52() strict Category-like V65() V66() V67() with_identities ) CatStr ) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
= F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) holds
opp ((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
(b3 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
(opp p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(opp p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
opp (F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) opp) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let I be ( ( ) ( )
set ) ;
let F be ( (
Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
let f be ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ;
func F * f -> ( (
Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
means
for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) holds
it : ( (
Function-like ) (
Relation-like K20(
I : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined I : ( ( ) ( )
set )
-valued Function-like )
Element of
K19(
K20(
K20(
I : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= (F : ( ( Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like I : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
(*) f : ( (
Function-like V18(
I : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
I : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ;
func f * F -> ( (
Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
means
for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) holds
it : ( (
Function-like ) (
Relation-like K20(
I : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined I : ( ( ) ( )
set )
-valued Function-like )
Element of
K19(
K20(
K20(
I : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= f : ( (
Function-like V18(
I : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
I : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
(*) (F : ( ( Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like I : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
p1,
p2,
f being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
* f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
(p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) (*) f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) (*) f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
f,
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
* ((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
(f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) (*) p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) (*) p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
f being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) st
doms F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= I : ( ( ) ( )
set )
--> (cod f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(
doms (F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) * f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
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carrier of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
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set ) )
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carrier of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
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set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
cods (F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) * f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
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-defined the
carrier of
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void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
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carrier of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
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carrier of
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void Category-like V65()
V66()
V67()
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empty non
void V52()
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V66()
V67()
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Category) : ( ( ) ( non
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= cods F : ( (
Function-like V18(
b1 : ( ( ) ( )
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carrier' of
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void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
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set ) ) ) (
Relation-like b1 : ( ( ) ( )
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-defined the
carrier' of
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void Category-like V65()
V66()
V67()
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empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
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carrier' of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
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carrier of
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void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
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carrier of
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void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
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set ) ) )
Function of
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carrier of
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void Category-like V65()
V66()
V67()
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empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
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theorem
for
I being ( ( ) ( )
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for
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void Category-like V65()
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void V52()
Category-like V65()
V66()
V67()
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Category)
for
f being ( ( ) ( )
Morphism of ( ( ) ( non
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Function-like V18(
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void Category-like V65()
V66()
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empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
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-defined the
carrier' of
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void Category-like V65()
V66()
V67()
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empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
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carrier' of
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empty non
void Category-like V65()
V66()
V67()
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empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
empty )
set ) ) st
cods F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
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empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= I : ( ( ) ( )
set )
--> (dom f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
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Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
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(
doms (f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) * F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= doms F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
cods (f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) * F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= I : ( ( ) ( )
set )
--> (cod f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let I be ( ( ) ( )
set ) ;
let F,
G be ( (
Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
func F "*" G -> ( (
Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
means
for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) holds
it : ( (
Function-like ) (
Relation-like K20(
I : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined I : ( ( ) ( )
set )
-valued Function-like )
Element of
K19(
K20(
K20(
I : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
I : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= (F : ( ( Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like I : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
(*) (G : ( ( Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like I : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(I : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
F,
G being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) st
doms F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cods G : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
(
doms (F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) "*" G : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= doms G : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
cods (F : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) "*" G : ( ( Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cods F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
p1,
p2,
q1,
q2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
"*" ((x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) ) --> (q1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ,q2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) )) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
(p1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) (*) q1 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(p2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) (*) q2 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
f being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
* f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
"*" (I : ( ( ) ( ) set ) --> f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
f being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
* F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (I : ( ( ) ( ) set ) --> f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
"*" F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
begin
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let a,
b be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
let IT be ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ;
attr IT is
retraction means
(
Hom (
a : ( ( ) ( )
set ) ,
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) & ex
g being ( ( ) ( )
Morphism of
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
a : ( ( ) ( )
set ) ) st
IT : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
* g : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )
= id b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) );
attr IT is
coretraction means
(
Hom (
a : ( ( ) ( )
set ) ,
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) & ex
g being ( ( ) ( )
Morphism of
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
a : ( ( ) ( )
set ) ) st
g : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* IT : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) )
= id a : ( ( ) ( )
set ) : ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) );
end;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c,
a,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
Hom (
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
g : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
retraction holds
g : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
retraction ;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a,
b,
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
coretraction holds
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
coretraction ;
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let a,
b be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
assume
Hom (
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set )
;
let D be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let T be ( ( ) (
Relation-like the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
D : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
D : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ) ;
let f be ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ;
func T /. f -> ( ( ) ( )
Morphism of
T : ( (
Function-like ) (
Relation-like K20(
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined a : ( ( ) ( )
set )
-valued Function-like )
Element of
K19(
K20(
K20(
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. a : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier of
D : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
T : ( (
Function-like ) (
Relation-like K20(
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined a : ( ( ) ( )
set )
-valued Function-like )
Element of
K19(
K20(
K20(
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
D : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )
equals
T : ( (
Function-like ) (
Relation-like K20(
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-defined a : ( ( ) ( )
set )
-valued Function-like )
Element of
K19(
K20(
K20(
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. f : ( ( ) ( )
Morphism of
b : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ,
D : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
C,
D being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
D : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ) st
f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
retraction holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) )
/. f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b6 : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) )
. b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
b6 : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) )
. b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) is
retraction ;
theorem
for
C,
D being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
D : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ) st
f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
coretraction holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) )
/. f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b6 : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) )
. b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
b6 : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) )
. b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) is
coretraction ;
begin
begin
begin
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let a be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
let I be ( ( ) ( )
set ) ;
mode Projections_family of
a,
I -> ( (
Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
means
doms it : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
--> a : ( ( ) ( )
set ) : ( (
Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Element of
K19(
K20(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
dom p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
dom p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) holds
(
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is ( ( ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
{x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
for
G being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) st
doms F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cods G : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
"*" G : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
f being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
cod f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) holds
(f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) opp) : ( ( ) ( )
Element of the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
* (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Projections_family of cod b3 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) opp) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
= (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Projections_family of cod b3 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) * f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
opp : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ;
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let a be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
let I be ( ( ) ( )
set ) ;
let F be ( (
Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
pred a is_a_product_wrt F means
(
F : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is ( ( ) (
Relation-like I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Projections_family of
a : ( ( ) ( )
set ) ,
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
b being ( ( ) ( )
Object of ( ( ) ( )
set ) )
for
F9 being ( ( ) (
Relation-like I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Projections_family of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) st
cods F : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= cods F9 : ( ( ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b1 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) : ( (
Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) holds
ex
h being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
(
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
k being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
( ( for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( (
Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
a : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
(F : ( ( Function-like V18(a : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like a : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(a : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(a : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
(*) k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= F9 : ( ( ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b1 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) iff
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
= k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) ) ) ) );
end;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c,
d being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) )
for
F9 being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) st
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) &
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt F9 : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) &
cods F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cods F9 : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
are_isomorphic ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) st
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) & ( for
x1,
x2 being ( ( ) ( )
set ) st
x1 : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) &
x2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
in I : ( ( ) ( )
set ) holds
Hom (
(cod (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Projections_family of b3 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(cod (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Projections_family of b3 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) ) holds
for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) holds
for
d being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) st
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
= cod (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Projections_family of b3 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
Hom (
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
for
f being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b6 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b6 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
retraction ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) st
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) holds
for
f being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
dom f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
cod f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
invertible holds
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) )
* f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let c be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
let p1,
p2 be ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ;
pred c is_a_product_wrt p1,
p2 means
(
dom p1 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= c : ( ( ) ( )
set ) &
dom p2 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= c : ( ( ) ( )
set ) & ( for
d being ( ( ) ( )
Object of ( ( ) ( )
set ) )
for
f,
g being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
(cod p1 : ( ( Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like c : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
g : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
(cod p2 : ( ( Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like c : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
ex
h being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
(
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
k being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
( (
p1 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
(*) k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) &
p2 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
(*) k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= g : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) iff
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
= k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) ) ) ) );
end;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
(
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) iff
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c,
a,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) st
Hom (
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
for
p1 being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
p2 being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) holds
(
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt p1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ,
p2 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) iff for
d being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) st
Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
(
Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) & ( for
f being ( ( ) ( )
Morphism of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
g being ( ( ) ( )
Morphism of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ex
h being ( ( ) ( )
Morphism of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
for
k being ( ( ) ( )
Morphism of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) holds
( (
p1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* k : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= f : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) &
p2 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* k : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= g : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ) iff
h : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= k : ( ( ) ( )
Morphism of
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ) ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c,
a,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
p1 being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
p2 being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
Hom (
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt p1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ,
p2 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) &
Hom (
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
(
p1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
retraction &
p2 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
retraction ) ;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
d,
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
for
f being ( ( ) ( )
Morphism of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) &
dom f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
cod f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
invertible holds
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
(*) f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
(*) f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
begin
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let c be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
let I be ( ( ) ( )
set ) ;
mode Injections_family of
c,
I -> ( (
Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
means
cods it : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
--> c : ( ( ) ( )
set ) : ( (
Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Element of
K19(
K20(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
p1,
p2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
cod p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
cod p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) holds
(
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
p1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
p2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is ( ( ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
{x1 : ( ( ) ( ) set ) ,x2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) )
for
G being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) st
doms F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cods G : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) )
"*" G : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
(
F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) iff
F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
opp : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
opp : ( ( ) ( )
Element of the
carrier of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
(C : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) holds
(
F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) iff
opp F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Projections_family of
opp c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
f being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
dom f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) holds
(F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Injections_family of dom b3 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) opp) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
* (f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) opp) : ( ( ) ( )
Element of the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
= (f : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) * F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Injections_family of dom b3 : ( ( ) ( ) Morphism of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
opp : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ;
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let c be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
let I be ( ( ) ( )
set ) ;
let F be ( (
Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
pred c is_a_coproduct_wrt F means
(
F : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) is ( ( ) (
Relation-like I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Injections_family of
c : ( ( ) ( )
set ) ,
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
d being ( ( ) ( )
Object of ( ( ) ( )
set ) )
for
F9 being ( ( ) (
Relation-like I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Injections_family of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) st
doms F : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= doms F9 : ( ( ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b1 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) : ( (
Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set )
-valued Function-like V18(
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) )
Function of
I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) , the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) holds
ex
h being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
(
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
c : ( ( ) ( )
set ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
k being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
c : ( ( ) ( )
set ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
( ( for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
(*) (F : ( ( Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like c : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= F9 : ( ( ) (
Relation-like I : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b1 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ) iff
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
= k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) ) ) ) );
end;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
I : ( ( ) ( )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
(
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_product_wrt F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) iff
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
opp : ( ( ) ( )
Element of the
carrier of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt F : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
opp : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
(b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) opp) : ( ( non
empty non
void strict ) ( non
empty non
void V52()
strict Category-like V65()
V66()
V67()
with_identities )
CatStr ) : ( ( ) ( non
empty )
set ) ) ) ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c,
d being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) )
for
F9 being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) st
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) &
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt F9 : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) &
doms F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= doms F9 : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
are_isomorphic ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) st
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) & ( for
x1,
x2 being ( ( ) ( )
set ) st
x1 : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) &
x2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
in I : ( ( ) ( )
set ) holds
Hom (
(dom (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Injections_family of b3 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) /. x1 : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(dom (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Injections_family of b3 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) /. x2 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) ) holds
for
x being ( ( ) ( )
set ) st
x : ( ( ) ( )
set )
in I : ( ( ) ( )
set ) holds
for
d being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) st
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
= dom (F : ( ( ) ( Relation-like b1 : ( ( ) ( ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like V18(b1 : ( ( ) ( ) set ) , the carrier' of b2 : ( ( non empty non void Category-like V65() V66() V67() with_identities ) ( non empty non void V52() Category-like V65() V66() V67() with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Injections_family of b3 : ( ( ) ( ) Object of ( ( ) ( non empty ) set ) ) ,b1 : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) set ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
Hom (
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
for
f being ( ( ) ( )
Morphism of
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
f : ( ( ) ( )
Morphism of
b6 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) )
/. x : ( ( ) ( )
set ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
f : ( ( ) ( )
Morphism of
b6 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
coretraction ;
theorem
for
I being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
F being ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
I : ( ( ) ( )
set ) ) st
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) &
dom f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
cod f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
invertible holds
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* F : ( ( ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Injections_family of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
set ) ) : ( (
Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( ) ( )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of
b1 : ( ( ) ( )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
definition
let C be ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) ;
let c be ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ;
let i1,
i2 be ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ;
pred c is_a_coproduct_wrt i1,
i2 means
(
cod i1 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= c : ( ( ) ( )
set ) &
cod i2 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= c : ( ( ) ( )
set ) & ( for
d being ( ( ) ( )
Object of ( ( ) ( )
set ) )
for
f,
g being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
(dom i1 : ( ( Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like c : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
g : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
(dom i2 : ( ( Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like c : ( ( ) ( ) set ) -defined C : ( ( non empty ) ( non empty ) set ) -valued Function-like V18(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ) Element of K19(K20(c : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
ex
h being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
(
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
c : ( ( ) ( )
set ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) & ( for
k being ( ( ) ( )
Morphism of ( ( ) ( )
set ) ) st
k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
in Hom (
c : ( ( ) ( )
set ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) holds
( (
k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
(*) i1 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= f : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) &
k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
(*) i2 : ( (
Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like c : ( ( ) ( )
set )
-defined C : ( ( non
empty ) ( non
empty )
set )
-valued Function-like V18(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K19(
K20(
c : ( ( ) ( )
set ) ,
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty ) ( non
empty )
set ) : ( ( ) ( )
set ) )
= g : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) iff
h : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
= k : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) ) ) ) );
end;
theorem
for
x1,
x2 being ( ( ) ( )
set )
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
i1,
i2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) st
x1 : ( ( ) ( )
set )
<> x2 : ( ( ) ( )
set ) holds
(
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt i1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
i2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) iff
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt (
x1 : ( ( ) ( )
set ) ,
x2 : ( ( ) ( )
set ) )
--> (
i1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
i2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ) : ( (
Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like {b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like V18(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
K19(
K20(
{b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) } : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a,
c,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) st
Hom (
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
for
i1 being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
i2 being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) holds
(
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt i1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ,
i2 : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) iff for
d being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) st
Hom (
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
(
Hom (
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) & ( for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ex
h being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
for
k being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) holds
( (
k : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* i1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) &
k : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
* i2 : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= g : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ) iff
h : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
= k : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b7 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ) ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
a,
c,
b being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
i1 being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
for
i2 being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
Hom (
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt i1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) ,
i2 : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) &
Hom (
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) &
Hom (
b : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of
K19( the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
<> {} : ( ( ) (
empty )
set ) holds
(
i1 : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
coretraction &
i2 : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
coretraction ) ;
theorem
for
C being ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category)
for
c,
d being ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
for
i1,
i2 being ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) )
for
f being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) st
c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt i1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) ,
i2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) &
dom f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= c : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
cod f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) &
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ) is
invertible holds
d : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) )
is_a_coproduct_wrt f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
(*) i1 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) ,
b3 : ( ( ) ( )
Object of ( ( ) ( non
empty )
set ) ) )
(*) i2 : ( ( ) ( )
Morphism of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier' of
b1 : ( ( non
empty non
void Category-like V65()
V66()
V67()
with_identities ) ( non
empty non
void V52()
Category-like V65()
V66()
V67()
with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;