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

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