
theorem R3: :: REAL_3:3
  for a,b be Integer holds a mod b = b*frac (a/b)
  proof
    let a,b be Integer;
    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 - (a div b)*b by INT_1:def 10
      .= a - [\a/b/]*b by INT_1:def 9
      .= a/b*b - [\a/b/]*b by XCMPLX_1:87
      .= (a/b - [\a/b/])*b
      .= b*(frac (a/b)) by INT_1:def 8;
      hence thesis;
    end;
  end;
