
theorem Th44:
  for C being with_binary_products category, a,b,c,d being Object of C
  st Hom(a,b) <> {} & Hom(c,d) <> {} holds Hom(a [x] c,b [x] d)<> {}
  proof
    let C be with_binary_products category;
    let a,b,c,d be Object of C;
    assume
A1: Hom(a,b) <> {};
    assume
A2: Hom(c,d) <> {};
A3: a [x] c, pr1(a,c), pr2(a,c) is_product_of a,c &
    Hom(a [x] c,a) <> {} & Hom(a [x] c,c) <> {} by Th42;
A4: b [x] d, pr1(b,d), pr2(b,d) is_product_of b,d &
    Hom(b [x] d,b) <> {} & Hom(b [x] d,d) <> {} by Th42;
    Hom(a [x] c,b) <> {} & Hom(a [x] c,d) <> {} by A3,A1,A2,CAT_7:22;
    hence thesis by A4,Def10;
  end;
