
theorem
  311 is prime
proof
  now
    311 = 2*155 + 1; hence not 2 divides 311 by NAT_4:9;
    311 = 3*103 + 2; hence not 3 divides 311 by NAT_4:9;
    311 = 5*62 + 1; hence not 5 divides 311 by NAT_4:9;
    311 = 7*44 + 3; hence not 7 divides 311 by NAT_4:9;
    311 = 11*28 + 3; hence not 11 divides 311 by NAT_4:9;
    311 = 13*23 + 12; hence not 13 divides 311 by NAT_4:9;
    311 = 17*18 + 5; hence not 17 divides 311 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 311 & n is prime
  holds not n divides 311 by XPRIMET1:14;
  hence thesis by NAT_4:14;
