
theorem
  for a be Integer, b be even Integer st
    not b divides a holds a mod b is odd implies (-a) mod b is odd
  proof
    let a be Integer, b be even Integer such that
    A1: not b divides a;
    per cases;
    suppose b is zero;
      hence thesis;
    end;
    suppose not b is zero; then
      reconsider b as non zero Integer;
      a mod b + ((-a) mod b) is even by A1,MOD;
      hence thesis;
    end;
  end;
