
theorem EVP:
  for a,b be Integer holds  (a+b) is even iff parity a = parity b
  proof
    let a,b be Integer;
    thus (a+b) is even implies parity a = parity b
    proof
      assume (a+b) is even; then
      0 = parity (a+b)
      .= |.(parity a)-(parity b).| by ABP; then
      (parity a) - (parity b) = 0;
      hence thesis;
    end;
    assume parity a = parity b; then
    0 = |.(parity a) - (parity b).|
    .= parity (a+b) by ABP;
    hence thesis;
  end;
