
theorem Lemma4:
  for A being 2-element set, a, b being Element of A st a <> b holds
    id A = {[a, a], [b, b]}
  proof
    let A be 2-element set, a, b be Element of A;
    assume
A0: a <> b;
    set c = a, d = b;
    set x1 = [a,a], x2 = [b,b];
T0: A = {a, b} by Lemma3, A0;
T2: id {a,b,c,d} = id {a,a,b,b} by ENUMSET1:62
      .= id {a,b,b} by ENUMSET1:31
      .= id {b,b,a} by ENUMSET1:59
      .= id {a,b} by ENUMSET1:30;
    {[a,a],[b,b],[c,c],[d,d]} = {x1, x1, x2, x2} by ENUMSET1:62
      .= {x1, x2, x2} by ENUMSET1:31
      .= {x2, x2, x1} by ENUMSET1:59
      .= {x1, x2} by ENUMSET1:30;
    hence thesis by NECKLA_3:2,T2,T0;
  end;
