
theorem Th4:
  for i2,i1 being Integer st i2 < 0 holds -(i1 mod i2) < -i2
  proof
    let i2,i1 be Integer;
    assume A1: i2 <0;
    (i1 / i2) - 1 < [\(i1 / i2)/] by INT_1:def 6;
    then (i1 div i2) * i2 < ((i1 / i2) - 1) * i2 by A1,XREAL_1:69;
    then (i1 div i2) * i2 < ((i1 / i2) * i2) - (1 * i2);
    then ((i1 div i2) * i2) < i1 - i2 by A1,XCMPLX_1:87;
    then ((i1 div i2) * i2) -i1 < i1 -i2 - i1 by XREAL_1:14;
    then -(i1- ((i1 div i2) * i2) ) < -i2;
    hence -( i1 mod i2) < -i2 by A1,INT_1:def 10;
  end;
