begin
theorem
for
C being ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr )
for
a,
b,
c,
d being ( ( ) ( )
Object of
C : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ,
d : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
are_equipotent &
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
= {f : ( ( ) ( ) Morphism of b2 : ( ( ) ( ) Object of b1 : ( ( non empty non void ) ( non empty non void V59() ) CatStr ) ) ,b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void ) ( non empty non void V59() ) CatStr ) ) ) } : ( ( ) ( non
empty trivial )
set ) holds
ex
h being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ,
d : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ) st
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ,
d : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
= {h : ( ( ) ( ) Morphism of b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void ) ( non empty non void V59() ) CatStr ) ) ,b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void ) ( non empty non void V59() ) CatStr ) ) ) } : ( ( ) ( non
empty trivial )
set ) ;
definition
let C be ( ( non
empty non
void ) ( non
empty non
void V59() )
CatStr ) ;
attr C is
with_identities means
for
a being ( ( ) ( )
Element of ( ( ) ( )
set ) ) ex
i being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) st
for
b being ( ( ) ( )
Element of ( ( ) ( )
set ) ) holds
( (
Hom (
a : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) implies for
g being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) holds
g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) )
(*) i : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
C : ( ( ) ( )
MultiGraphStruct ) )
= g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) ) & (
Hom (
b : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) implies for
f being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
a : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) holds
i : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) )
(*) f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Morphism of
C : ( ( ) ( )
MultiGraphStruct ) )
= f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) ) );
end;
definition
let o,
m be ( ( ) ( )
set ) ;
func 1Cat (
o,
m)
-> ( (
strict ) (
strict )
CatStr )
equals
CatStr(#
{o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non
empty trivial )
set ) ,
{m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set ) ,
(m : ( ( ) ( ) set ) :-> o : ( ( ) ( ) MultiGraphStruct ) ) : ( (
Function-like V25(
{m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set ) ,
{o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non
empty trivial )
set ) ) ) (
Relation-like {m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set )
-defined {o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non
empty trivial )
set )
-valued Function-like non
empty total V25(
{m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set ) ,
{o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non
empty trivial )
set ) ) )
Element of
bool [:{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) ,{o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
(m : ( ( ) ( ) set ) :-> o : ( ( ) ( ) MultiGraphStruct ) ) : ( (
Function-like V25(
{m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set ) ,
{o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non
empty trivial )
set ) ) ) (
Relation-like {m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set )
-defined {o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non
empty trivial )
set )
-valued Function-like non
empty total V25(
{m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set ) ,
{o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non
empty trivial )
set ) ) )
Element of
bool [:{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) ,{o : ( ( ) ( ) MultiGraphStruct ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
((m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) :-> m : ( ( ) ( ) set ) ) : ( (
Function-like V25(
[:{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) ,{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) (
Relation-like )
set ) ,
{m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set ) ) ) (
Relation-like [:{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) ,{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) (
Relation-like )
set )
-defined {m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set )
-valued Function-like total V25(
[:{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) ,{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) (
Relation-like )
set ) ,
{m : ( ( ) ( ) set ) } : ( ( ) ( non
empty trivial )
set ) ) )
Element of
bool [:[:{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) ,{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) ( Relation-like ) set ) ,{m : ( ( ) ( ) set ) } : ( ( ) ( non empty trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) #) : ( (
strict ) (
strict )
CatStr ) ;
end;
registration
let o,
m be ( ( ) ( )
set ) ;
end;
registration
let o,
m be ( ( ) ( )
set ) ;
end;
definition
let C be ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ;
let a be ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ;
func id a -> ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
a : ( ( ) ( )
set ) )
means
for
b being ( ( ) ( )
Object of
C : ( ( ) ( )
set ) ) holds
( (
Hom (
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) implies for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ) holds
f : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ,
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) )
(*) it : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
C : ( ( ) ( )
set ) )
= f : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ,
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ) ) & (
Hom (
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) implies for
f being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ,
a : ( ( ) ( )
set ) ) holds
it : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
(*) f : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ,
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ) : ( ( ) ( )
Morphism of
C : ( ( ) ( )
set ) )
= f : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ,
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void reflexive with_identities ) ( non
empty non
void V59()
reflexive with_identities )
CatStr ) ) ) ) );
end;
theorem
for
o,
m being ( ( ) ( )
set )
for
a,
b,
c,
d being ( ( ) ( )
Object of
(1Cat (o : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) ,
b : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) )
for
g being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) ,
d : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) ) holds
f : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) ,
b4 : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) )
= g : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) ,
b6 : ( ( ) ( )
Object of
(1Cat (b1 : ( ( ) ( ) set ) ,b2 : ( ( ) ( ) set ) )) : ( (
strict ) ( non
empty trivial V53() non
void 1 : ( ( ) ( non
empty )
set )
-element V59()
trivial' strict Category-like transitive associative reflexive with_identities )
CatStr ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b,
c,
d being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
for
h being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
d : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
d : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
(h : ( ( ) ( ) Morphism of b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) * g : ( ( ) ( ) Morphism of b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= h : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* (g : ( ( ) ( ) Morphism of b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) * f : ( ( ) ( ) Morphism of b2 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
definition
let C be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let b,
c be ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
let g be ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
attr g is
monic means
(
Hom (
b : ( ( ) ( )
set ) ,
c : ( (
Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like b : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:b : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ( for
a being ( ( ) ( )
Object of
C : ( ( ) ( )
set ) ) st
Hom (
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
set ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
f1,
f2 being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
set ) ) st
g : ( (
Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like b : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:b : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
* f1 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( (
Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like b : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:b : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) )
= g : ( (
Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like b : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:b : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
* f2 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( (
Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like b : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
b : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:b : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ) holds
f1 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= f2 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) );
end;
definition
let C be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let a,
b be ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
let f be ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
attr f is
epi means
(
Hom (
a : ( ( ) ( )
set ) ,
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ( for
c being ( ( ) ( )
Object of
C : ( ( ) ( )
set ) ) st
Hom (
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
g1,
g2 being ( ( ) ( )
Morphism of
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
g1 : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= g2 : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
g1 : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= g2 : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) );
end;
definition
let C be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let a,
b be ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
let f be ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
attr f is
invertible means
(
Hom (
a : ( ( ) ( )
set ) ,
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ex
g being ( ( ) ( )
Morphism of
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
a : ( ( ) ( )
set ) ) st
(
f : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
* g : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) )
= id b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ) &
g : ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
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 transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
b,
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
(
g : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
monic iff for
a being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f1,
f2 being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
g : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f1 : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= g : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f2 : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
f1 : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= f2 : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
b,
c,
d being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
for
h being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
d : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
d : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
h : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* g : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
monic holds
g : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
monic ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
h being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
h : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
monic ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
(
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
epi iff for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
g1,
g2 being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
g1 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= g2 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
g1 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= g2 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b,
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
epi holds
g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
epi ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
h being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
h : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
h : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
epi ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
(
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
invertible iff (
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ex
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
(
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) &
g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) ) ) ;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
g1,
g2 being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* g1 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) &
g2 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
g1 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= g2 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
definition
let C be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let a,
b be ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
let f be ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
assume
f : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
invertible
;
func f " -> ( ( ) ( )
Morphism of
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
a : ( ( ) ( )
set ) )
means
(
f : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
* it : ( ( ) ( )
Element of
C : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) )
= id b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
b : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ) &
it : ( ( ) ( )
Element of
C : ( ( ) ( )
set ) )
* f : ( (
Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like a : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
a : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:a : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
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 transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b,
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
invertible &
g : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) is
invertible holds
(g : ( ( ) ( ) Morphism of b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) * f : ( ( ) ( ) Morphism of b2 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
" : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= (f : ( ( ) ( ) Morphism of b2 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ") : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* (g : ( ( ) ( ) Morphism of b3 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ") : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
definition
let C be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let a be ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
attr a is
terminal means
for
b being ( ( ) ( )
Object of
C : ( ( ) ( )
set ) ) holds
(
Hom (
b : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ,
a : ( ( ) ( )
set ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ex
f being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ,
a : ( ( ) ( )
set ) ) st
for
g being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ,
a : ( ( ) ( )
set ) ) holds
f : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) )
= g : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) );
attr a is
initial means
for
b being ( ( ) ( )
Object of
C : ( ( ) ( )
set ) ) holds
(
Hom (
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ex
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) st
for
g being ( ( ) ( )
Morphism of
a : ( ( ) ( )
set ) ,
b : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) holds
f : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) )
= g : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b1 : ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) );
let b be ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
reflexivity
for a being ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ex f being ( ( ) ( ) Morphism of a : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ,a : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) st f : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b1 : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) is invertible
;
symmetry
for a, b being ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) st ex f being ( ( ) ( ) Morphism of a : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ,b : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b1 : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) ) st f : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b1 : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) is invertible holds
ex f being ( ( ) ( ) Morphism of b : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b1 : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) ,a : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) st f : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b1 : ( ( ) ( ) Morphism of a : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,b : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) ) is invertible
;
end;
theorem
for
C being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
a,
b being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
are_isomorphic iff (
Hom (
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) &
Hom (
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ex
f being ( ( ) ( )
Morphism of
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ex
f9 being ( ( ) ( )
Morphism of
b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
(
f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f9 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) &
f9 : ( ( ) ( )
Morphism of
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
* f : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= id a : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) ) ) ;
definition
let C,
D be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
mode Functor of
C,
D -> ( (
Function-like V25( the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set )
-defined the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set )
-valued Function-like non
empty total V25( the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) )
Function of the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
means
( ( for
c being ( ( ) ( )
Element of ( ( ) ( )
set ) ) ex
d being ( ( ) ( )
Element of ( ( ) ( )
set ) ) st
it : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. (id c : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
= id d : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ) & ( for
f being ( ( ) ( )
Element of the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) holds
(
it : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. (id (dom f : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of C : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Morphism of
dom b1 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ,
dom b1 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
= id (dom (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . f : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier' of D : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
dom (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . b1 : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ,
dom (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . b1 : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ) &
it : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. (id (cod f : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of C : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Morphism of
cod b1 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ,
cod b1 : ( ( ) ( )
Element of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
= id (cod (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . f : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier' of D : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
cod (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . b1 : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ,
cod (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . b1 : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ) ) ) & ( for
f,
g being ( ( ) ( )
Element of the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) st
[g : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ,f : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ] : ( ( ) ( non
empty )
set )
in dom the
Comp of
C : ( ( ) ( )
set ) : ( (
Function-like ) (
Relation-like [: the carrier' of C : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier' of C : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set )
-defined the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set )
-valued Function-like )
PartFunc of ,) : ( ( ) (
Relation-like )
set ) holds
it : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. (g : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) (*) f : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
C : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
= (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . g : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
(*) (it : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . f : ( ( ) ( ) Element of the carrier' of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Morphism of
D : ( ( ) ( )
set ) ) ) );
end;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) st ( for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ex
d being ( ( ) ( )
Object of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (id c : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id d : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) & ( for
f being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (id (dom f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
dom b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
dom b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id (dom (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
dom (b3 : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
dom (b3 : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) &
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (id (cod f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
cod b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
cod b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id (cod (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
cod (b3 : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
cod (b3 : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ) ) & ( for
f,
g being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
dom g : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cod f : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (g : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) (*) f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . g : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
(*) (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (id (dom f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
dom b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
dom b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id (dom (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
dom (b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
dom (b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) &
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (id (cod f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
cod b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
cod b4 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id (cod (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( ) Element of the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
cod (b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
cod (b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f,
g being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
dom g : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cod f : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
(
dom (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . g : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cod (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (g : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) (*) f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . g : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
(*) (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
for
F being ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) st ( for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (id c : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id (F : ( ( Function-like V25( the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . c : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
b4 : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. b5 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
b4 : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. b5 : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ) & ( for
f being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
F : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (dom f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= dom (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
F : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (cod f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cod (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ) & ( for
f,
g being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
dom g : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cod f : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) holds
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (g : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) (*) f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . g : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
(*) (T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) is ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
definition
let C,
D be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let F be ( (
Function-like V25( the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
assume
for
c being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ex
d being ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) st
F : ( (
Function-like V25( the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (id c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id d : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) )
;
func Obj F -> ( (
Function-like V25( the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set )
-defined the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set )
-valued Function-like non
empty total V25( the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) )
Function of the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
means
for
c being ( ( ) ( )
Element of ( ( ) ( )
set ) )
for
d being ( ( ) ( )
Element of ( ( ) ( )
set ) ) st
F : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. (id c : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b1 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
= id d : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
b2 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ,
b2 : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ) holds
it : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. c : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
= d : ( ( ) ( )
Element of ( ( ) ( non
empty )
set ) ) ;
end;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) st ( for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ex
d being ( ( ) ( )
Object of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (id c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id d : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ) holds
for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
d being ( ( ) ( )
Object of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
T : ( (
Function-like V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (id c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id d : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
(Obj T : ( ( Function-like V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= d : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
d being ( ( ) ( )
Object of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (id c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id d : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b5 : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
(Obj T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= d : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (id c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id ((Obj T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( Function-like V25( the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Morphism of
(Obj b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ,
(Obj b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
(Obj T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (dom f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= dom (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
(Obj T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. (cod f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= cod (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (id c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= id (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Morphism of
b3 : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b3 : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
Morphism of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (dom f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
= dom (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) &
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (cod f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
= cod (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . f : ( ( ) ( ) Morphism of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) ;
theorem
for
B,
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
S being ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
S : ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
* T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( (
Function-like ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Element of
bool [: the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) is ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
definition
let B,
C,
D be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let T be ( ( ) (
Relation-like the
carrier' of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
let S be ( ( ) (
Relation-like the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
*redefine func S * T -> ( ( ) (
Relation-like the
carrier' of
B : ( ( ) ( )
set ) : ( ( ) ( )
set )
-defined the
carrier' of
D : ( (
Function-like V25(
C : ( ( ) ( )
set ) ,
B : ( ( ) ( )
set ) ) ) (
Relation-like C : ( ( ) ( )
set )
-defined B : ( ( ) ( )
set )
-valued Function-like V25(
C : ( ( ) ( )
set ) ,
B : ( ( ) ( )
set ) ) )
Element of
bool [:C : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
-valued Function-like non
empty total V25( the
carrier' of
B : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier' of
D : ( (
Function-like V25(
C : ( ( ) ( )
set ) ,
B : ( ( ) ( )
set ) ) ) (
Relation-like C : ( ( ) ( )
set )
-defined B : ( ( ) ( )
set )
-valued Function-like V25(
C : ( ( ) ( )
set ) ,
B : ( ( ) ( )
set ) ) )
Element of
bool [:C : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )
Functor of
B : ( ( ) ( )
set ) ,
D : ( (
Function-like V25(
C : ( ( ) ( )
set ) ,
B : ( ( ) ( )
set ) ) ) (
Relation-like C : ( ( ) ( )
set )
-defined B : ( ( ) ( )
set )
-valued Function-like V25(
C : ( ( ) ( )
set ) ,
B : ( ( ) ( )
set ) ) )
Element of
bool [:C : ( ( ) ( ) set ) ,B : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ) ;
end;
theorem
for
B,
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
S being ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
b being ( ( ) ( )
Object of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(Obj (S : ( ( ) ( Relation-like the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) * T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( (
Function-like V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
= (Obj S : ( ( ) ( Relation-like the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( (
Function-like V25( the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ) (
Relation-like the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Function of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
. ((Obj T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( Function-like V25( the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Function of the carrier of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) . b : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) : ( ( ) ( )
Element of the
carrier of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
B,
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
S being ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
b being ( ( ) ( )
Object of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(S : ( ( ) ( Relation-like the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b3 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) * T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. b : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
= S : ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
definition
let C,
D be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let T be ( ( ) (
Relation-like the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
attr T is
isomorphic means
(
T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) is
one-to-one &
rng T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set )
= the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) &
rng (Obj T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ) : ( (
Function-like V25( the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ) (
Relation-like the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set )
-defined the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set )
-valued Function-like non
empty total V25( the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) )
Function of the
carrier of
C : ( ( ) ( )
set ) : ( ( ) ( )
set ) , the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) : ( ( ) ( non
empty )
set )
= the
carrier of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) );
attr T is
full means
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( ) ( )
set ) ) st
Hom (
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ,
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c9 : ( ( ) ( ) Object of C : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
g being ( ( ) ( )
Morphism of
T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ,
T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. c9 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ) holds
(
Hom (
c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) & ex
f being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
g : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. f : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) );
attr T is
faithful means
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( ) ( )
set ) ) st
Hom (
c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
f1,
f2 being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) st
T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. f1 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) )
= T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
. f2 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) holds
f1 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) )
= f2 : ( ( ) ( )
Morphism of
b1 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b2 : ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
end;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
f being ( ( ) ( )
set ) st
f : ( ( ) ( )
set )
in Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. f : ( ( ) ( )
set ) : ( ( ) ( )
set )
in Hom (
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
f being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) )
in Hom (
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
f being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) is ( ( ) ( )
Morphism of
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
Hom (
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) ;
theorem
for
B,
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
S being ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
full &
S : ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
full holds
S : ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
* T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
full ;
theorem
for
B,
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
B : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
S being ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
faithful &
S : ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
faithful holds
S : ( ( ) (
Relation-like the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
* T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b3 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
faithful ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
.: (Hom (c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) )) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) : ( ( ) ( )
set )
c= Hom (
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ;
definition
let C,
D be ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ;
let T be ( ( ) (
Relation-like the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
let c,
c9 be ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ;
func hom (
T,
c,
c9)
-> ( (
Function-like V25(
Hom (
c : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
c9 : ( ( ) ( )
Element of
C : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ,
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c9 : ( ( ) ( ) Element of C : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) ) (
Relation-like Hom (
c : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
c9 : ( ( ) ( )
Element of
C : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-defined Hom (
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ,
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c9 : ( ( ) ( ) Element of C : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-valued Function-like V25(
Hom (
c : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
c9 : ( ( ) ( )
Element of
C : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ,
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c9 : ( ( ) ( ) Element of C : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) )
Function of
Hom (
c : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
c9 : ( ( ) ( )
Element of
C : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ,
(T : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) . c9 : ( ( ) ( ) Element of C : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Object of
D : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) )
equals
T : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) )
| (Hom (c : ( ( Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) ( Relation-like D : ( ( ) ( ) set ) -defined C : ( ( ) ( ) set ) -valued Function-like V25(D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) ) ) Element of bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) ) ,c9 : ( ( ) ( ) Element of C : ( ( ) ( ) set ) ) )) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) : ( (
Function-like ) (
Relation-like Hom (
c : ( (
Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) ) (
Relation-like D : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued Function-like V25(
D : ( ( ) ( )
set ) ,
C : ( ( ) ( )
set ) ) )
Element of
bool [:D : ( ( ) ( ) set ) ,C : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ,
c9 : ( ( ) ( )
Element of
C : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-defined D : ( ( ) ( )
set )
-defined the
carrier' of
C : ( ( ) ( )
set ) : ( ( ) ( )
set )
-defined C : ( ( ) ( )
set )
-valued the
carrier' of
D : ( ( ) ( )
set ) : ( ( ) ( )
set )
-valued Function-like )
Element of
bool [: the carrier' of C : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier' of D : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) ( )
set ) ) ;
end;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) st
Hom (
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
<> {} : ( ( ) (
Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty trivial )
set ) holds
for
f being ( ( ) ( )
Morphism of
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) holds
(hom (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) )) : ( (
Function-like V25(
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) ) (
Relation-like Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-defined Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-valued Function-like V25(
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) )
Function of
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) )
. f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
set )
= T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) )
. f : ( ( ) ( )
Morphism of
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Element of the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
full iff for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
rng (hom (T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ,c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) )) : ( (
Function-like V25(
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) ) (
Relation-like Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-defined Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-valued Function-like V25(
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) )
Function of
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) ( )
set )
= Hom (
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(T : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . c9 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) ;
theorem
for
C,
D being ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category)
for
T being ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
D : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
(
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) is
faithful iff for
c,
c9 being ( ( ) ( )
Object of
C : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) holds
hom (
T : ( ( ) (
Relation-like the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-defined the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set )
-valued Function-like non
empty total V25( the
carrier' of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) , the
carrier' of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) : ( ( ) ( non
empty )
set ) ) )
Functor of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ,
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
c9 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( (
Function-like V25(
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) ) (
Relation-like Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-defined Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) )
-valued Function-like V25(
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) )
Function of
Hom (
b4 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
b5 : ( ( ) ( )
Object of
b1 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ,
Hom (
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b4 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ,
(b3 : ( ( ) ( Relation-like the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -defined the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) -valued Function-like non empty total V25( the carrier' of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) , the carrier' of b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) : ( ( ) ( non empty ) set ) ) ) Functor of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ,b2 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) . b5 : ( ( ) ( ) Object of b1 : ( ( non empty non void Category-like transitive associative reflexive with_identities ) ( non empty non void V59() Category-like transitive associative reflexive with_identities ) Category) ) ) : ( ( ) ( )
Object of
b2 : ( ( non
empty non
void Category-like transitive associative reflexive with_identities ) ( non
empty non
void V59()
Category-like transitive associative reflexive with_identities )
Category) ) ) : ( ( ) ( )
Subset of ( ( ) ( )
set ) ) ) is
one-to-one ) ;