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

theorem Th41:
  for f being Morphism of a,b, h being Morphism of c,d, g being
  Morphism of e,a, 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:>
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 e,c;
  assume that
A1: Hom(a,b) <> {} and
A2: Hom(c,d) <> {} and
A3: Hom(e,a) <> {} & Hom(e,c) <> {};
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 thesis by A4,A8,A6,A9,Th25;
end;
