let C be Cartesian_category; :: thesis: for a, b, c, d, e being Object of C
for f being Morphism of a,b
for h being Morphism of c,d
for g being Morphism of e,a
for k being Morphism of e,c st Hom (a,b) <> {} & Hom (c,d) <> {} & Hom (e,a) <> {} & Hom (e,c) <> {} holds
(f [x] h) * <:g,k:> = <:(f * g),(h * k):>
let a, b, c, d, e be Object of C; :: thesis: for f being Morphism of a,b
for h being Morphism of c,d
for g being Morphism of e,a
for k being Morphism of e,c st Hom (a,b) <> {} & Hom (c,d) <> {} & Hom (e,a) <> {} & Hom (e,c) <> {} holds
(f [x] h) * <:g,k:> = <:(f * g),(h * k):>
let f be Morphism of a,b; :: thesis: for h being Morphism of c,d
for g being Morphism of e,a
for k being Morphism of e,c st Hom (a,b) <> {} & Hom (c,d) <> {} & Hom (e,a) <> {} & Hom (e,c) <> {} holds
(f [x] h) * <:g,k:> = <:(f * g),(h * k):>
let h be Morphism of c,d; :: thesis: for g being Morphism of e,a
for k being Morphism of e,c st Hom (a,b) <> {} & Hom (c,d) <> {} & Hom (e,a) <> {} & Hom (e,c) <> {} holds
(f [x] h) * <:g,k:> = <:(f * g),(h * k):>
let g be Morphism of e,a; :: thesis: for k being Morphism of e,c st Hom (a,b) <> {} & Hom (c,d) <> {} & Hom (e,a) <> {} & Hom (e,c) <> {} holds
(f [x] h) * <:g,k:> = <:(f * g),(h * k):>
let k be Morphism of e,c; :: thesis: ( Hom (a,b) <> {} & Hom (c,d) <> {} & Hom (e,a) <> {} & Hom (e,c) <> {} implies (f [x] h) * <:g,k:> = <:(f * g),(h * k):> )
assume that
A1: Hom (a,b) <> {} and
A2: Hom (c,d) <> {} and
A3: ( Hom (e,a) <> {} & Hom (e,c) <> {} ) ; :: thesis: (f [x] h) * <:g,k:> = <:(f * g),(h * k):>
A4: Hom (e,(a [x] c)) <> {} by A3, Th23;
A5: Hom ((a [x] c),c) <> {} by Th19;
then A6: Hom ((a [x] c),d) <> {} by A2, CAT_1:24;
A7: Hom ((a [x] c),a) <> {} by Th19;
then A8: Hom ((a [x] c),b) <> {} by A1, CAT_1:24;
(pr2 (a,c)) * <:g,k:> = k by A3, Def10;
then A9: h * k = (h * (pr2 (a,c))) * <:g,k:> by A2, A4, A5, CAT_1:25;
(pr1 (a,c)) * <:g,k:> = g by A3, Def10;
then f * g = (f * (pr1 (a,c))) * <:g,k:> by A1, A4, A7, CAT_1:25;
hence (f [x] h) * <:g,k:> = <:(f * g),(h * k):> by A4, A8, A6, A9, Th25; :: thesis: verum