let I be set ; :: thesis: for C being Category
for f being Morphism of C
for F being Function of I, the carrier' of C st cods F = I --> (dom f) holds
( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )

let C be Category; :: thesis: for f being Morphism of C
for F being Function of I, the carrier' of C st cods F = I --> (dom f) holds
( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )

let f be Morphism of C; :: thesis: for F being Function of I, the carrier' of C st cods F = I --> (dom f) holds
( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )

let F be Function of I, the carrier' of C; :: thesis: ( cods F = I --> (dom f) implies ( doms (f * F) = doms F & cods (f * F) = I --> (cod f) ) )
assume A1: cods F = I --> (dom f) ; :: thesis: ( doms (f * F) = doms F & cods (f * F) = I --> (cod f) )
now :: thesis: for x being set st x in I holds
(doms F) /. x = (doms (f * F)) /. x
let x be set ; :: thesis: ( x in I implies (doms F) /. x = (doms (f * F)) /. x )
assume A2: x in I ; :: thesis: (doms F) /. x = (doms (f * F)) /. x
then A3: cod (F /. x) = (I --> (dom f)) /. x by
.= dom f by ;
thus (doms F) /. x = dom (F /. x) by
.= dom (f (*) (F /. x)) by
.= dom ((f * F) /. x) by
.= (doms (f * F)) /. x by ; :: thesis: verum
end;
hence doms (f * F) = doms F by Th1; :: thesis: cods (f * F) = I --> (cod f)
now :: thesis: for x being set st x in I holds
(cods (f * F)) /. x = (I --> (cod f)) /. x
let x be set ; :: thesis: ( x in I implies (cods (f * F)) /. x = (I --> (cod f)) /. x )
assume A4: x in I ; :: thesis: (cods (f * F)) /. x = (I --> (cod f)) /. x
then A5: cod (F /. x) = (I --> (dom f)) /. x by
.= dom f by ;
thus (cods (f * F)) /. x = cod ((f * F) /. x) by
.= cod (f (*) (F /. x)) by
.= cod f by
.= (I --> (cod f)) /. x by ; :: thesis: verum
end;
hence cods (f * F) = I --> (cod f) by Th1; :: thesis: verum