
theorem
  317 is prime
proof
  now
    317 = 2*158 + 1; hence not 2 divides 317 by NAT_4:9;
    317 = 3*105 + 2; hence not 3 divides 317 by NAT_4:9;
    317 = 5*63 + 2; hence not 5 divides 317 by NAT_4:9;
    317 = 7*45 + 2; hence not 7 divides 317 by NAT_4:9;
    317 = 11*28 + 9; hence not 11 divides 317 by NAT_4:9;
    317 = 13*24 + 5; hence not 13 divides 317 by NAT_4:9;
    317 = 17*18 + 11; hence not 17 divides 317 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 317 & n is prime
  holds not n divides 317 by XPRIMET1:14;
  hence thesis by NAT_4:14;
