
theorem
  for p be Prime, n be Nat holds (n choose p) mod p = (n div p) mod p
  proof
    let p be Prime, n be Nat;
    (1*p + 0) mod p = 0 mod p &
      (1*p + 0) div p = (0 div p) + 1 by NAT_D:61; then
    (n choose p) mod p = (((n mod p) choose 0)*((n div p) choose 1)) mod p
      by AL
    .= (1*(n div p)) mod p by NEWTON:19;
    hence thesis;
  end;
