reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th1:
  k <> 0 & i < l & l <= j & k divides l implies i div k < j div k
proof
  assume that
A1: k <> 0 and
A2: i < l and
A3: l <= j and
A4: k divides l;
A5: l mod k = 0 by A1,A4,PEPIN:6;
  i + (i mod k) >= i by NAT_1:11;
  then i - (i mod k) <= (i + (i mod k)) - (i mod k) by XREAL_1:9;
  then i - (i mod k) < l by A2,XXREAL_0:2;
  then
A6: (i - (i mod k)) / k < l / k by A1,XREAL_1:74;
  i = k * (i div k) + (i mod k) by A1,NAT_D:2;
  then k / k * (i div k) = (i - (i mod k)) / k;
  then
A7: 1 * (i div k) = (i - (i mod k)) / k by A1,XCMPLX_1:60;
  l = k * (l div k) + (l mod k) by A1,NAT_D:2
    .= k * (l div k) by A5;
  then k / k * (l div k) = l / k;
  then
A8: 1 * (l div k) = l / k by A1,XCMPLX_1:60;
  l div k <= j div k by A3,NAT_2:24;
  hence thesis by A8,A7,A6,XXREAL_0:2;
end;
