
theorem NPK:
  for n,k be Nat, p be Prime st k < p holds
    ((n*p + k) choose k) mod p = 1
  proof
    let n,k be Nat, p be Prime such that
    A1: k < p;
    ((n*p + k) choose k) mod p =
      (((n*p + k) mod p) choose k) mod p by A1,MOC
    .= ((k mod p) choose k) mod p by NAT_D:61
    .= (k choose k) mod p by A1,MOC
    .= 1 mod (1 + (p - 1)) by NEWTON:21;
    hence thesis;
  end;
