
theorem
  for i, j being Integer st j > 0 & i div j = 0 holds i < j
proof
  let i, j be Integer;
  assume that
A1: j > 0 and
A2: i div j = 0;
  [\ i / j /] = 0 by A2,INT_1:def 9;
  then i/j - 1 < 0 by INT_1:def 6;
  then i/j < 0+1 by XREAL_1:19;
  then i/j*j < 1*j by A1,XREAL_1:68;
  hence thesis by A1,XCMPLX_1:87;
end;
