
theorem
  313 is prime
proof
  now
    313 = 2*156 + 1; hence not 2 divides 313 by NAT_4:9;
    313 = 3*104 + 1; hence not 3 divides 313 by NAT_4:9;
    313 = 5*62 + 3; hence not 5 divides 313 by NAT_4:9;
    313 = 7*44 + 5; hence not 7 divides 313 by NAT_4:9;
    313 = 11*28 + 5; hence not 11 divides 313 by NAT_4:9;
    313 = 13*24 + 1; hence not 13 divides 313 by NAT_4:9;
    313 = 17*18 + 7; hence not 17 divides 313 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 313 & n is prime
  holds not n divides 313 by XPRIMET1:14;
  hence thesis by NAT_4:14;
end;
