
theorem
  2309 is prime
proof
  now
    2309 = 2*1154 + 1; hence not 2 divides 2309 by NAT_4:9;
    2309 = 3*769 + 2; hence not 3 divides 2309 by NAT_4:9;
    2309 = 5*461 + 4; hence not 5 divides 2309 by NAT_4:9;
    2309 = 7*329 + 6; hence not 7 divides 2309 by NAT_4:9;
    2309 = 11*209 + 10; hence not 11 divides 2309 by NAT_4:9;
    2309 = 13*177 + 8; hence not 13 divides 2309 by NAT_4:9;
    2309 = 17*135 + 14; hence not 17 divides 2309 by NAT_4:9;
    2309 = 19*121 + 10; hence not 19 divides 2309 by NAT_4:9;
    2309 = 23*100 + 9; hence not 23 divides 2309 by NAT_4:9;
    2309 = 29*79 + 18; hence not 29 divides 2309 by NAT_4:9;
    2309 = 31*74 + 15; hence not 31 divides 2309 by NAT_4:9;
    2309 = 37*62 + 15; hence not 37 divides 2309 by NAT_4:9;
    2309 = 41*56 + 13; hence not 41 divides 2309 by NAT_4:9;
    2309 = 43*53 + 30; hence not 43 divides 2309 by NAT_4:9;
    2309 = 47*49 + 6; hence not 47 divides 2309 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 2309 & n is prime
  holds not n divides 2309 by XPRIMET1:30;
  hence thesis by NAT_4:14;
end;
