
theorem
  8219 is prime
proof
  now
    8219 = 2*4109 + 1; hence not 2 divides 8219 by NAT_4:9;
    8219 = 3*2739 + 2; hence not 3 divides 8219 by NAT_4:9;
    8219 = 5*1643 + 4; hence not 5 divides 8219 by NAT_4:9;
    8219 = 7*1174 + 1; hence not 7 divides 8219 by NAT_4:9;
    8219 = 11*747 + 2; hence not 11 divides 8219 by NAT_4:9;
    8219 = 13*632 + 3; hence not 13 divides 8219 by NAT_4:9;
    8219 = 17*483 + 8; hence not 17 divides 8219 by NAT_4:9;
    8219 = 19*432 + 11; hence not 19 divides 8219 by NAT_4:9;
    8219 = 23*357 + 8; hence not 23 divides 8219 by NAT_4:9;
    8219 = 29*283 + 12; hence not 29 divides 8219 by NAT_4:9;
    8219 = 31*265 + 4; hence not 31 divides 8219 by NAT_4:9;
    8219 = 37*222 + 5; hence not 37 divides 8219 by NAT_4:9;
    8219 = 41*200 + 19; hence not 41 divides 8219 by NAT_4:9;
    8219 = 43*191 + 6; hence not 43 divides 8219 by NAT_4:9;
    8219 = 47*174 + 41; hence not 47 divides 8219 by NAT_4:9;
    8219 = 53*155 + 4; hence not 53 divides 8219 by NAT_4:9;
    8219 = 59*139 + 18; hence not 59 divides 8219 by NAT_4:9;
    8219 = 61*134 + 45; hence not 61 divides 8219 by NAT_4:9;
    8219 = 67*122 + 45; hence not 67 divides 8219 by NAT_4:9;
    8219 = 71*115 + 54; hence not 71 divides 8219 by NAT_4:9;
    8219 = 73*112 + 43; hence not 73 divides 8219 by NAT_4:9;
    8219 = 79*104 + 3; hence not 79 divides 8219 by NAT_4:9;
    8219 = 83*99 + 2; hence not 83 divides 8219 by NAT_4:9;
    8219 = 89*92 + 31; hence not 89 divides 8219 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8219 & n is prime
  holds not n divides 8219 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
