
theorem NPR:
  for n be Nat, m be non zero Nat holds
    (n mod m) choose (m - 1) = 1 iff n mod m = m - 1
  proof
    let n be Nat, m be non zero Nat;
    n mod m <> m - 1 implies (n mod m) choose (m - 1) = 0
    proof
      assume
      B1: n mod m <> m - 1;
      n mod m < (m - 1) + 1 by NAT_D:1; then
      n mod m <= (m - 1) by INT_1:7; then
      n mod m < m - 1 by B1,XXREAL_0:1;
      hence thesis by NEWTON:def 3;
    end;
    hence thesis by NEWTON:21;
  end;
