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

theorem Th1:
  for i,j be Integer st j < 0 holds j < i mod j <= 0
  proof
    let x,j be Integer;
    assume A1: j < 0;
    then A2: x/j*j = x by XCMPLX_1:87;
    x/j -1 < [\ x/j /] by INT_1:def 6;
    then x/j -1 < (x div j) by INT_1:def 9;
    then (x/j -1)*j > (x div j)*j by A1,XREAL_1:69;
    then x -j > (x div j)*j-0 by A2;
    then x -(x div j)*j > j-0 by XREAL_1:16;
    hence x mod j > j by INT_1:def 10,A1;
    [\ x/j /] <= x/j by INT_1:def 6;
    then (x div j) <= x/j by INT_1:def 9;
    then (x div j)*j >= x/j*j by A1,XREAL_1:65;
    then 0 >= x -(x div j)*j by A2,XREAL_1:47;
    hence x mod j <=0  by INT_1:def 10;
  end;
