
theorem
  8669 is prime
proof
  now
    8669 = 2*4334 + 1; hence not 2 divides 8669 by NAT_4:9;
    8669 = 3*2889 + 2; hence not 3 divides 8669 by NAT_4:9;
    8669 = 5*1733 + 4; hence not 5 divides 8669 by NAT_4:9;
    8669 = 7*1238 + 3; hence not 7 divides 8669 by NAT_4:9;
    8669 = 11*788 + 1; hence not 11 divides 8669 by NAT_4:9;
    8669 = 13*666 + 11; hence not 13 divides 8669 by NAT_4:9;
    8669 = 17*509 + 16; hence not 17 divides 8669 by NAT_4:9;
    8669 = 19*456 + 5; hence not 19 divides 8669 by NAT_4:9;
    8669 = 23*376 + 21; hence not 23 divides 8669 by NAT_4:9;
    8669 = 29*298 + 27; hence not 29 divides 8669 by NAT_4:9;
    8669 = 31*279 + 20; hence not 31 divides 8669 by NAT_4:9;
    8669 = 37*234 + 11; hence not 37 divides 8669 by NAT_4:9;
    8669 = 41*211 + 18; hence not 41 divides 8669 by NAT_4:9;
    8669 = 43*201 + 26; hence not 43 divides 8669 by NAT_4:9;
    8669 = 47*184 + 21; hence not 47 divides 8669 by NAT_4:9;
    8669 = 53*163 + 30; hence not 53 divides 8669 by NAT_4:9;
    8669 = 59*146 + 55; hence not 59 divides 8669 by NAT_4:9;
    8669 = 61*142 + 7; hence not 61 divides 8669 by NAT_4:9;
    8669 = 67*129 + 26; hence not 67 divides 8669 by NAT_4:9;
    8669 = 71*122 + 7; hence not 71 divides 8669 by NAT_4:9;
    8669 = 73*118 + 55; hence not 73 divides 8669 by NAT_4:9;
    8669 = 79*109 + 58; hence not 79 divides 8669 by NAT_4:9;
    8669 = 83*104 + 37; hence not 83 divides 8669 by NAT_4:9;
    8669 = 89*97 + 36; hence not 89 divides 8669 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8669 & n is prime
  holds not n divides 8669 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
