
theorem Th3:
  for i2,i1 being Integer st i2 <= 0 holds i1 mod i2 <= 0
  proof
    let i2,i1 be Integer;
    assume A1: i2 <= 0;
    per cases by A1;
    suppose A2: i2 < 0;
      [\(i1 / i2)/] <= i1 / i2 by INT_1:def 6;
      then (i1 / i2) * i2 <= (i1 div i2) * i2 by A2,XREAL_1:65;
      then i1 <= (i1 div i2) * i2 by A2,XCMPLX_1:87;
      then i1 - ((i1 div i2) * i2) <= 0 by XREAL_1:47;
      hence i1 mod i2 <= 0 by INT_1:def 10;
    end;
    suppose i2 = 0;
      hence i1 mod i2 <= 0 by INT_1:def 10;
    end;
  end;
