
theorem for a be Integer, n be non zero Nat holds parity (a|^n) = parity a
  proof
    let a be Integer, n be non zero Nat;
    per cases;
    suppose
      A1: a is even; then
      a|^n is even;
      hence thesis by A1;
    end;
    suppose
      A1: a is odd; then
      a|^n is odd;
      hence thesis by A1;
    end;
  end;
