
theorem NPP:
  for n be Nat, m be non zero Nat holds (n mod m) choose (m - 1) < 2
  proof
    let n be Nat, m be non zero Nat;
    n mod m < (m - 1) + 1 by NAT_D:1; then
    n mod m <= (m - 1) by INT_1:7; then
    per cases by XXREAL_0:1;
    suppose
      n mod m = (m - 1); then
      (n mod m) choose (m - 1) = 1 by NEWTON:21;
      hence thesis;
    end;
    suppose
      n mod m < m - 1;
      hence thesis by NEWTON:def 3;
    end;
  end;
