
theorem
  409 is prime
proof
  now
    409 = 2*204 + 1; hence not 2 divides 409 by NAT_4:9;
    409 = 3*136 + 1; hence not 3 divides 409 by NAT_4:9;
    409 = 5*81 + 4; hence not 5 divides 409 by NAT_4:9;
    409 = 7*58 + 3; hence not 7 divides 409 by NAT_4:9;
    409 = 11*37 + 2; hence not 11 divides 409 by NAT_4:9;
    409 = 13*31 + 6; hence not 13 divides 409 by NAT_4:9;
    409 = 17*24 + 1; hence not 17 divides 409 by NAT_4:9;
    409 = 19*21 + 10; hence not 19 divides 409 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 409 & n is prime
  holds not n divides 409 by XPRIMET1:16;
  hence thesis by NAT_4:14;
