
theorem
  9311 is prime
proof
  now
    9311 = 2*4655 + 1; hence not 2 divides 9311 by NAT_4:9;
    9311 = 3*3103 + 2; hence not 3 divides 9311 by NAT_4:9;
    9311 = 5*1862 + 1; hence not 5 divides 9311 by NAT_4:9;
    9311 = 7*1330 + 1; hence not 7 divides 9311 by NAT_4:9;
    9311 = 11*846 + 5; hence not 11 divides 9311 by NAT_4:9;
    9311 = 13*716 + 3; hence not 13 divides 9311 by NAT_4:9;
    9311 = 17*547 + 12; hence not 17 divides 9311 by NAT_4:9;
    9311 = 19*490 + 1; hence not 19 divides 9311 by NAT_4:9;
    9311 = 23*404 + 19; hence not 23 divides 9311 by NAT_4:9;
    9311 = 29*321 + 2; hence not 29 divides 9311 by NAT_4:9;
    9311 = 31*300 + 11; hence not 31 divides 9311 by NAT_4:9;
    9311 = 37*251 + 24; hence not 37 divides 9311 by NAT_4:9;
    9311 = 41*227 + 4; hence not 41 divides 9311 by NAT_4:9;
    9311 = 43*216 + 23; hence not 43 divides 9311 by NAT_4:9;
    9311 = 47*198 + 5; hence not 47 divides 9311 by NAT_4:9;
    9311 = 53*175 + 36; hence not 53 divides 9311 by NAT_4:9;
    9311 = 59*157 + 48; hence not 59 divides 9311 by NAT_4:9;
    9311 = 61*152 + 39; hence not 61 divides 9311 by NAT_4:9;
    9311 = 67*138 + 65; hence not 67 divides 9311 by NAT_4:9;
    9311 = 71*131 + 10; hence not 71 divides 9311 by NAT_4:9;
    9311 = 73*127 + 40; hence not 73 divides 9311 by NAT_4:9;
    9311 = 79*117 + 68; hence not 79 divides 9311 by NAT_4:9;
    9311 = 83*112 + 15; hence not 83 divides 9311 by NAT_4:9;
    9311 = 89*104 + 55; hence not 89 divides 9311 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 9311 & n is prime
  holds not n divides 9311 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
