
theorem ILP: for a,b be Integer holds Parity (a*b) = (Parity a)*(Parity b)
  proof
    let a,b be Integer;
    per cases;
    suppose a = 0 or b = 0; then
      (Parity a = 0 or Parity b = 0) & Parity (a*b) = 0 by Def1;
      hence thesis;
    end;
    suppose
      B1: a <> 0 & b <> 0; then
      B2: Parity b = 2|^(2|-count b) & Parity a = 2|^(2|-count a) by Def1;
      Parity (a*b) = 2|^(2|-count (a*b)) by B1,Def1
      .= 2|^((2|-count a) + (2|-count b)) by B1,NEWTON03:57,INT_2:28
      .= (2|^(2|-count a)) * (2|^(2|-count b)) by NEWTON:8;
      hence thesis by B2;
    end;
  end;
