reserve o,m for set;
reserve C for Cartesian_category;
reserve a,b,c,d,e,s for Object of C;

theorem
  for f being Morphism of a,b, h being Morphism of c,d, g being Morphism
  of e,a, 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)
proof
  let f be Morphism of a,b, h be Morphism of c,d;
  let g be Morphism of e,a, k be Morphism of s,c;
  assume that
A1: Hom(a,b) <> {} and
A2: Hom(c,d) <> {} and
A3: Hom(e,a) <> {} and
A4: Hom(s,c) <> {};
A5: Hom(e[x]s,s) <> {} by Th19;
  then
A6: Hom(e[x]s,c) <> {} by A4,CAT_1:24;
A7: Hom(e[x]s,e) <> {} by Th19;
  then f*(g*pr1(e,s)) = (f*g)*pr1(e,s) by A1,A3,CAT_1:25;
  then
A8: (f*g)[x](h*k) = <:f*(g*pr1(e,s)),h*(k*pr2(e,s)):> by A2,A4,A5,CAT_1:25;
  Hom(e[x]s,a) <> {} by A3,A7,CAT_1:24;
  hence thesis by A1,A2,A6,A8,Th41;
end;
