
theorem
  8009 is prime
proof
  now
    8009 = 2*4004 + 1; hence not 2 divides 8009 by NAT_4:9;
    8009 = 3*2669 + 2; hence not 3 divides 8009 by NAT_4:9;
    8009 = 5*1601 + 4; hence not 5 divides 8009 by NAT_4:9;
    8009 = 7*1144 + 1; hence not 7 divides 8009 by NAT_4:9;
    8009 = 11*728 + 1; hence not 11 divides 8009 by NAT_4:9;
    8009 = 13*616 + 1; hence not 13 divides 8009 by NAT_4:9;
    8009 = 17*471 + 2; hence not 17 divides 8009 by NAT_4:9;
    8009 = 19*421 + 10; hence not 19 divides 8009 by NAT_4:9;
    8009 = 23*348 + 5; hence not 23 divides 8009 by NAT_4:9;
    8009 = 29*276 + 5; hence not 29 divides 8009 by NAT_4:9;
    8009 = 31*258 + 11; hence not 31 divides 8009 by NAT_4:9;
    8009 = 37*216 + 17; hence not 37 divides 8009 by NAT_4:9;
    8009 = 41*195 + 14; hence not 41 divides 8009 by NAT_4:9;
    8009 = 43*186 + 11; hence not 43 divides 8009 by NAT_4:9;
    8009 = 47*170 + 19; hence not 47 divides 8009 by NAT_4:9;
    8009 = 53*151 + 6; hence not 53 divides 8009 by NAT_4:9;
    8009 = 59*135 + 44; hence not 59 divides 8009 by NAT_4:9;
    8009 = 61*131 + 18; hence not 61 divides 8009 by NAT_4:9;
    8009 = 67*119 + 36; hence not 67 divides 8009 by NAT_4:9;
    8009 = 71*112 + 57; hence not 71 divides 8009 by NAT_4:9;
    8009 = 73*109 + 52; hence not 73 divides 8009 by NAT_4:9;
    8009 = 79*101 + 30; hence not 79 divides 8009 by NAT_4:9;
    8009 = 83*96 + 41; hence not 83 divides 8009 by NAT_4:9;
    8009 = 89*89 + 88; hence not 89 divides 8009 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8009 & n is prime
  holds not n divides 8009 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
