
theorem
  for a,b be Integer holds
    (a mod b) + ((-a) mod b) = 0 or (a mod b) + ((-a) mod b) = b
  proof
    let a,b be Integer;
    per cases;
    suppose b is zero; then
      reconsider b as zero Integer;
      a mod b = 0 & (-a) mod b = 0;
      hence thesis;
    end;
    suppose not b is zero; then
      reconsider b as non zero Integer;
      per cases;
      suppose
        b divides a; then
        (a mod b) = 0 & ((-a) mod b ) = 0 by INT162,INT_2:10;
        hence thesis;
      end;
      suppose
        not b divides a;
        hence thesis by MOD;
      end;
    end;
  end;
