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