let C be Category; :: thesis: for g, f, g9, f9 being Morphism of C st cod g = dom f & dom g9 = cod f9 holds
hom ((f * g),(g9 * f9)) = (hom (g,g9)) * (hom (f,f9))
let g, f, g9, f9 be Morphism of C; :: thesis: ( cod g = dom f & dom g9 = cod f9 implies hom ((f * g),(g9 * f9)) = (hom (g,g9)) * (hom (f,f9)) )
assume that
A1: cod g = dom f and
A2: dom g9 = cod f9 ; :: thesis: hom ((f * g),(g9 * f9)) = (hom (g,g9)) * (hom (f,f9))
A3: dom (g9 * f9) = dom f9 by A2, CAT_1:42;
A4: cod (f * g) = cod f by A1, CAT_1:42;
A5: ( cod (g9 * f9) = cod g9 & dom (f * g) = dom g ) by A1, A2, CAT_1:42;

now

set h = hom ((f * g),(g9 * f9));
set h2 = hom (g,g9);
set h1 = hom (f,f9);
A6: ( Hom ((dom f),(cod f9)) = {} implies Hom ((cod f),(dom f9)) = {} ) by Th51;
A7: ( Hom ((dom g),(cod g9)) = {} implies Hom ((cod g),(dom g9)) = {} ) by Th51;
hence dom (hom ((f * g),(g9 * f9))) = Hom ((cod f),(dom f9)) by A1, A2, A3, A5, A4, A6, FUNCT_2:def 1; :: thesis: ( dom ((hom (g,g9)) * (hom (f,f9))) = Hom ((cod f),(dom f9)) & ( for x being set st x in Hom ((cod f),(dom f9)) holds
(hom ((f * g),(g9 * f9))) . x = ((hom (g,g9)) * (hom (f,f9))) . x ) )
thus A8: dom ((hom (g,g9)) * (hom (f,f9))) = Hom ((cod f),(dom f9)) by A1, A2, A7, A6, FUNCT_2:def 1; :: thesis: for x being set st x in Hom ((cod f),(dom f9)) holds
(hom ((f * g),(g9 * f9))) . x = ((hom (g,g9)) * (hom (f,f9))) . x
let x be set ; :: thesis: ( x in Hom ((cod f),(dom f9)) implies (hom ((f * g),(g9 * f9))) . x = ((hom (g,g9)) * (hom (f,f9))) . x )
assume A9: x in Hom ((cod f),(dom f9)) ; :: thesis: (hom ((f * g),(g9 * f9))) . x = ((hom (g,g9)) * (hom (f,f9))) . x
then reconsider k = x as Morphism of C ;
A10: (hom (f,f9)) . x in Hom ((cod g),(dom g9)) by A1, A2, A9, Th51, FUNCT_2:7;
then reconsider l = (hom (f,f9)) . x as Morphism of C ;
A11: dom k = cod f by A9, CAT_1:18;
then A12: cod (k * (f * g)) = cod k by A4, CAT_1:42;
A13: cod k = dom f9 by A9, CAT_1:18;
then A14: dom (f9 * k) = dom k by CAT_1:42;
then A15: dom ((f9 * k) * f) = dom f by A11, CAT_1:42;
cod (f9 * k) = cod f9 by A13, CAT_1:42;
then A16: cod ((f9 * k) * f) = cod f9 by A11, A14, CAT_1:42;
thus (hom ((f * g),(g9 * f9))) . x = ((g9 * f9) * k) * (f * g) by A3, A4, A9, Def24
.= (g9 * f9) * (k * (f * g)) by A3, A4, A13, A11, CAT_1:43
.= g9 * (f9 * (k * (f * g))) by A2, A13, A12, CAT_1:43
.= g9 * ((f9 * k) * (f * g)) by A4, A13, A11, CAT_1:43
.= g9 * (((f9 * k) * f) * g) by A1, A11, A14, CAT_1:43
.= (g9 * ((f9 * k) * f)) * g by A1, A2, A15, A16, CAT_1:43
.= (g9 * l) * g by A9, Def24
.= (hom (g,g9)) . l by A10, Def24
.= ((hom (g,g9)) * (hom (f,f9))) . x by A8, A9, FUNCT_1:22 ; :: thesis: verum

end;

hence hom ((f * g),(g9 * f9)) = (hom (g,g9)) * (hom (f,f9)) by FUNCT_1:9; :: thesis: verum