
theorem
  8623 is prime
proof
  now
    8623 = 2*4311 + 1; hence not 2 divides 8623 by NAT_4:9;
    8623 = 3*2874 + 1; hence not 3 divides 8623 by NAT_4:9;
    8623 = 5*1724 + 3; hence not 5 divides 8623 by NAT_4:9;
    8623 = 7*1231 + 6; hence not 7 divides 8623 by NAT_4:9;
    8623 = 11*783 + 10; hence not 11 divides 8623 by NAT_4:9;
    8623 = 13*663 + 4; hence not 13 divides 8623 by NAT_4:9;
    8623 = 17*507 + 4; hence not 17 divides 8623 by NAT_4:9;
    8623 = 19*453 + 16; hence not 19 divides 8623 by NAT_4:9;
    8623 = 23*374 + 21; hence not 23 divides 8623 by NAT_4:9;
    8623 = 29*297 + 10; hence not 29 divides 8623 by NAT_4:9;
    8623 = 31*278 + 5; hence not 31 divides 8623 by NAT_4:9;
    8623 = 37*233 + 2; hence not 37 divides 8623 by NAT_4:9;
    8623 = 41*210 + 13; hence not 41 divides 8623 by NAT_4:9;
    8623 = 43*200 + 23; hence not 43 divides 8623 by NAT_4:9;
    8623 = 47*183 + 22; hence not 47 divides 8623 by NAT_4:9;
    8623 = 53*162 + 37; hence not 53 divides 8623 by NAT_4:9;
    8623 = 59*146 + 9; hence not 59 divides 8623 by NAT_4:9;
    8623 = 61*141 + 22; hence not 61 divides 8623 by NAT_4:9;
    8623 = 67*128 + 47; hence not 67 divides 8623 by NAT_4:9;
    8623 = 71*121 + 32; hence not 71 divides 8623 by NAT_4:9;
    8623 = 73*118 + 9; hence not 73 divides 8623 by NAT_4:9;
    8623 = 79*109 + 12; hence not 79 divides 8623 by NAT_4:9;
    8623 = 83*103 + 74; hence not 83 divides 8623 by NAT_4:9;
    8623 = 89*96 + 79; hence not 89 divides 8623 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8623 & n is prime
  holds not n divides 8623 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
