
theorem
  8209 is prime
proof
  now
    8209 = 2*4104 + 1; hence not 2 divides 8209 by NAT_4:9;
    8209 = 3*2736 + 1; hence not 3 divides 8209 by NAT_4:9;
    8209 = 5*1641 + 4; hence not 5 divides 8209 by NAT_4:9;
    8209 = 7*1172 + 5; hence not 7 divides 8209 by NAT_4:9;
    8209 = 11*746 + 3; hence not 11 divides 8209 by NAT_4:9;
    8209 = 13*631 + 6; hence not 13 divides 8209 by NAT_4:9;
    8209 = 17*482 + 15; hence not 17 divides 8209 by NAT_4:9;
    8209 = 19*432 + 1; hence not 19 divides 8209 by NAT_4:9;
    8209 = 23*356 + 21; hence not 23 divides 8209 by NAT_4:9;
    8209 = 29*283 + 2; hence not 29 divides 8209 by NAT_4:9;
    8209 = 31*264 + 25; hence not 31 divides 8209 by NAT_4:9;
    8209 = 37*221 + 32; hence not 37 divides 8209 by NAT_4:9;
    8209 = 41*200 + 9; hence not 41 divides 8209 by NAT_4:9;
    8209 = 43*190 + 39; hence not 43 divides 8209 by NAT_4:9;
    8209 = 47*174 + 31; hence not 47 divides 8209 by NAT_4:9;
    8209 = 53*154 + 47; hence not 53 divides 8209 by NAT_4:9;
    8209 = 59*139 + 8; hence not 59 divides 8209 by NAT_4:9;
    8209 = 61*134 + 35; hence not 61 divides 8209 by NAT_4:9;
    8209 = 67*122 + 35; hence not 67 divides 8209 by NAT_4:9;
    8209 = 71*115 + 44; hence not 71 divides 8209 by NAT_4:9;
    8209 = 73*112 + 33; hence not 73 divides 8209 by NAT_4:9;
    8209 = 79*103 + 72; hence not 79 divides 8209 by NAT_4:9;
    8209 = 83*98 + 75; hence not 83 divides 8209 by NAT_4:9;
    8209 = 89*92 + 21; hence not 89 divides 8209 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8209 & n is prime
  holds not n divides 8209 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
