
theorem
  2311 is prime
proof
  now
    2311 = 2*1155 + 1; hence not 2 divides 2311 by NAT_4:9;
    2311 = 3*770 + 1; hence not 3 divides 2311 by NAT_4:9;
    2311 = 5*462 + 1; hence not 5 divides 2311 by NAT_4:9;
    2311 = 7*330 + 1; hence not 7 divides 2311 by NAT_4:9;
    2311 = 11*210 + 1; hence not 11 divides 2311 by NAT_4:9;
    2311 = 13*177 + 10; hence not 13 divides 2311 by NAT_4:9;
    2311 = 17*135 + 16; hence not 17 divides 2311 by NAT_4:9;
    2311 = 19*121 + 12; hence not 19 divides 2311 by NAT_4:9;
    2311 = 23*100 + 11; hence not 23 divides 2311 by NAT_4:9;
    2311 = 29*79 + 20; hence not 29 divides 2311 by NAT_4:9;
    2311 = 31*74 + 17; hence not 31 divides 2311 by NAT_4:9;
    2311 = 37*62 + 17; hence not 37 divides 2311 by NAT_4:9;
    2311 = 41*56 + 15; hence not 41 divides 2311 by NAT_4:9;
    2311 = 43*53 + 32; hence not 43 divides 2311 by NAT_4:9;
    2311 = 47*49 + 8; hence not 47 divides 2311 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 2311 & n is prime
  holds not n divides 2311 by XPRIMET1:30;
  hence thesis by NAT_4:14;
end;
