
theorem
  8419 is prime
proof
  now
    8419 = 2*4209 + 1; hence not 2 divides 8419 by NAT_4:9;
    8419 = 3*2806 + 1; hence not 3 divides 8419 by NAT_4:9;
    8419 = 5*1683 + 4; hence not 5 divides 8419 by NAT_4:9;
    8419 = 7*1202 + 5; hence not 7 divides 8419 by NAT_4:9;
    8419 = 11*765 + 4; hence not 11 divides 8419 by NAT_4:9;
    8419 = 13*647 + 8; hence not 13 divides 8419 by NAT_4:9;
    8419 = 17*495 + 4; hence not 17 divides 8419 by NAT_4:9;
    8419 = 19*443 + 2; hence not 19 divides 8419 by NAT_4:9;
    8419 = 23*366 + 1; hence not 23 divides 8419 by NAT_4:9;
    8419 = 29*290 + 9; hence not 29 divides 8419 by NAT_4:9;
    8419 = 31*271 + 18; hence not 31 divides 8419 by NAT_4:9;
    8419 = 37*227 + 20; hence not 37 divides 8419 by NAT_4:9;
    8419 = 41*205 + 14; hence not 41 divides 8419 by NAT_4:9;
    8419 = 43*195 + 34; hence not 43 divides 8419 by NAT_4:9;
    8419 = 47*179 + 6; hence not 47 divides 8419 by NAT_4:9;
    8419 = 53*158 + 45; hence not 53 divides 8419 by NAT_4:9;
    8419 = 59*142 + 41; hence not 59 divides 8419 by NAT_4:9;
    8419 = 61*138 + 1; hence not 61 divides 8419 by NAT_4:9;
    8419 = 67*125 + 44; hence not 67 divides 8419 by NAT_4:9;
    8419 = 71*118 + 41; hence not 71 divides 8419 by NAT_4:9;
    8419 = 73*115 + 24; hence not 73 divides 8419 by NAT_4:9;
    8419 = 79*106 + 45; hence not 79 divides 8419 by NAT_4:9;
    8419 = 83*101 + 36; hence not 83 divides 8419 by NAT_4:9;
    8419 = 89*94 + 53; hence not 89 divides 8419 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8419 & n is prime
  holds not n divides 8419 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
