
theorem Th67:
  for C being with_binary_products non empty category, a,b being Object of C
  holds id- a [x] id- b = id-(a [x] b)
  proof
    let C be with_binary_products non empty category;
    let a,b be Object of C;
A1: Hom(a [x] b,a) <> {} by Th42;
A2: Hom(a,a)<>{} & Hom(b,b)<>{};
A3: (id- a) * pr1(a,b) = pr1(a,b) by A1,CAT_7:18
    .= pr1(a,b) * id-(a [x] b) by A1,CAT_7:18;
A4: Hom(a [x] b,b) <> {} by Th42;
    (id- b) * pr2(a,b) = pr2(a,b) by A4,CAT_7:18
    .= pr2(a,b) * id-(a [x] b) by A4,CAT_7:18;
    hence id- a [x] id- b = id-(a [x] b) by A2,A3,Def16;
  end;
