
theorem
  for a be positive Real, n being Nat holds [\n*a/] >= n*[\a/]
  proof
    let a be positive Real;
    let n be Nat;
    per cases;
    suppose n = 0;
      hence thesis;
    end;
    suppose
A0:   n <> 0;
A1:   a - 1 <= [\a/] <= a & n*a - 1 <= [\n*a/] <= n*a by INT_1:def 6; then
A2:   (n*a - 1) + 1 <= [\n*a/] + 1 by XREAL_1:6;
      per cases by A1,XXREAL_0:1;
      suppose [\a/] = a;
        hence thesis;
      end;
      suppose [\a/] < a; then
        n*[\a/] < n*a by A0,XREAL_1:68; then
        n*[\a/] < [\n*a/] + 1 by A2,XXREAL_0:2;
        hence thesis by INT_1:7;
      end;
    end;
  end;
