
theorem Th42:
  for C being with_binary_products category, a,b being Object of C holds
  a [x] b, pr1(a,b), pr2(a,b) is_product_of a,b &
  Hom(a [x] b,a) <> {} & Hom(a [x] b,b) <> {}
  proof
    let C be with_binary_products category;
    let a,b be Object of C;
    set T = the categorical_product of a,b;
    consider c be Object of C, p1 be Morphism of c,a, p2 be Morphism of c,b
    such that
A1: T = [c,p1,p2] & Hom(c,a) <> {} & Hom(c,b) <> {} &
    c,p1,p2 is_product_of a,b by Def12;
    thus thesis by A1;
  end;
