let C be Cartesian_category; :: thesis: for a, b, c, d, e, s 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 s,c st Hom a,b <> {} & Hom c,d <> {} & Hom e,a <> {} & Hom s,c <> {} holds
(f [x] h) * (g [x] k) = (f * g) [x] (h * k)
let a, b, c, d, e, s 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 s,c st Hom a,b <> {} & Hom c,d <> {} & Hom e,a <> {} & Hom s,c <> {} holds
(f [x] h) * (g [x] k) = (f * g) [x] (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 s,c st Hom a,b <> {} & Hom c,d <> {} & Hom e,a <> {} & Hom s,c <> {} holds
(f [x] h) * (g [x] k) = (f * g) [x] (h * k)
let h be Morphism of c,d; :: thesis: for g being Morphism of e,a
for k being Morphism of s,c st Hom a,b <> {} & Hom c,d <> {} & Hom e,a <> {} & Hom s,c <> {} holds
(f [x] h) * (g [x] k) = (f * g) [x] (h * k)
let g be Morphism of e,a; :: thesis: for k being Morphism of s,c st Hom a,b <> {} & Hom c,d <> {} & Hom e,a <> {} & Hom s,c <> {} holds
(f [x] h) * (g [x] k) = (f * g) [x] (h * k)
let k be Morphism of s,c; :: thesis: ( Hom a,b <> {} & Hom c,d <> {} & Hom e,a <> {} & Hom s,c <> {} implies (f [x] h) * (g [x] k) = (f * g) [x] (h * k) )
assume that
A1: Hom a,b <> {} and
A2: Hom c,d <> {} and
A3: Hom e,a <> {} and
A4: Hom s,c <> {} ; :: thesis: (f [x] h) * (g [x] k) = (f * g) [x] (h * k)
A5: Hom (e [x] s),s <> {} by Th21;
then A6: Hom (e [x] s),c <> {} by A4, CAT_1:52;
A7: Hom (e [x] s),e <> {} by Th21;
then f * (g * (pr1 e,s)) = (f * g) * (pr1 e,s) by A1, A3, CAT_1:54;
then A8: (f * g) [x] (h * k) = <:(f * (g * (pr1 e,s))),(h * (k * (pr2 e,s))):> by A2, A4, A5, CAT_1:54;
Hom (e [x] s),a <> {} by A3, A7, CAT_1:52;
hence (f [x] h) * (g [x] k) = (f * g) [x] (h * k) by A1, A2, A6, A8, Th43; :: thesis: verum