
theorem
  8581 is prime
proof
  now
    8581 = 2*4290 + 1; hence not 2 divides 8581 by NAT_4:9;
    8581 = 3*2860 + 1; hence not 3 divides 8581 by NAT_4:9;
    8581 = 5*1716 + 1; hence not 5 divides 8581 by NAT_4:9;
    8581 = 7*1225 + 6; hence not 7 divides 8581 by NAT_4:9;
    8581 = 11*780 + 1; hence not 11 divides 8581 by NAT_4:9;
    8581 = 13*660 + 1; hence not 13 divides 8581 by NAT_4:9;
    8581 = 17*504 + 13; hence not 17 divides 8581 by NAT_4:9;
    8581 = 19*451 + 12; hence not 19 divides 8581 by NAT_4:9;
    8581 = 23*373 + 2; hence not 23 divides 8581 by NAT_4:9;
    8581 = 29*295 + 26; hence not 29 divides 8581 by NAT_4:9;
    8581 = 31*276 + 25; hence not 31 divides 8581 by NAT_4:9;
    8581 = 37*231 + 34; hence not 37 divides 8581 by NAT_4:9;
    8581 = 41*209 + 12; hence not 41 divides 8581 by NAT_4:9;
    8581 = 43*199 + 24; hence not 43 divides 8581 by NAT_4:9;
    8581 = 47*182 + 27; hence not 47 divides 8581 by NAT_4:9;
    8581 = 53*161 + 48; hence not 53 divides 8581 by NAT_4:9;
    8581 = 59*145 + 26; hence not 59 divides 8581 by NAT_4:9;
    8581 = 61*140 + 41; hence not 61 divides 8581 by NAT_4:9;
    8581 = 67*128 + 5; hence not 67 divides 8581 by NAT_4:9;
    8581 = 71*120 + 61; hence not 71 divides 8581 by NAT_4:9;
    8581 = 73*117 + 40; hence not 73 divides 8581 by NAT_4:9;
    8581 = 79*108 + 49; hence not 79 divides 8581 by NAT_4:9;
    8581 = 83*103 + 32; hence not 83 divides 8581 by NAT_4:9;
    8581 = 89*96 + 37; hence not 89 divides 8581 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8581 & n is prime
  holds not n divides 8581 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
