
theorem
  for a be odd Nat holds Parity (a-1) = 2 * Parity(a div 2)
  proof
    let a be odd Nat;
    a = (a div 2)*2 + (a mod 2) by INT_1:59
    .= (a div 2)*2 + 1 by NAT_D:12; then
    Parity (a-1) = (Parity (a div 2))*Parity 2 by ILP;
    hence thesis;
  end;
