reserve n,n1,n2,k,D for Nat,
        r,r1,r2 for Real,
        x,y for Integer;

theorem Th2:
  for i,j be Integer st j <> 0 holds |. i mod j .| < |.j.|
  proof
    let x,j be Integer;
    assume j<>0; then
    per cases;
    suppose j > 0;
      then 0 <= x mod j < j & |.j.|= j by INT_1:57,58,ABSVALUE:def 1;
      hence thesis by ABSVALUE:def 1;
    end;
    suppose j < 0;
      then A2: j < x mod j <= 0 & |.j.| = -j by Th1,ABSVALUE:def 1;
      then |. x mod j .| = -(x mod j) by ABSVALUE:30;
      hence thesis by A2,XREAL_1:24;
    end;
  end;
