
theorem DIM:
  for a be non zero Integer, b be Integer holds
    a divides b iff a divides (b mod a)
  proof
    let a be non zero Integer, b be Integer;
    A1: a divides b implies a divides (b mod a)
    proof
      assume
      a divides b; then
      b mod a = 0 by INT162;
      hence thesis by INT_2:12;
    end;
    a divides (b mod a) implies a divides b
    proof
      assume
      B1: a divides (b mod a);
      per cases;
      suppose b is zero;
        hence thesis by INT_2:12;
      end;
      suppose
        not b is zero; then
        reconsider b as non zero Integer;
        |.b mod a.| < |.a.| by PELLS_EQ:2; then
        b mod a = 0 by B1,INT_4:6;
        hence thesis by INT162;
      end;
    end;
    hence thesis by A1;
  end;
