
theorem
  8501 is prime
proof
  now
    8501 = 2*4250 + 1; hence not 2 divides 8501 by NAT_4:9;
    8501 = 3*2833 + 2; hence not 3 divides 8501 by NAT_4:9;
    8501 = 5*1700 + 1; hence not 5 divides 8501 by NAT_4:9;
    8501 = 7*1214 + 3; hence not 7 divides 8501 by NAT_4:9;
    8501 = 11*772 + 9; hence not 11 divides 8501 by NAT_4:9;
    8501 = 13*653 + 12; hence not 13 divides 8501 by NAT_4:9;
    8501 = 17*500 + 1; hence not 17 divides 8501 by NAT_4:9;
    8501 = 19*447 + 8; hence not 19 divides 8501 by NAT_4:9;
    8501 = 23*369 + 14; hence not 23 divides 8501 by NAT_4:9;
    8501 = 29*293 + 4; hence not 29 divides 8501 by NAT_4:9;
    8501 = 31*274 + 7; hence not 31 divides 8501 by NAT_4:9;
    8501 = 37*229 + 28; hence not 37 divides 8501 by NAT_4:9;
    8501 = 41*207 + 14; hence not 41 divides 8501 by NAT_4:9;
    8501 = 43*197 + 30; hence not 43 divides 8501 by NAT_4:9;
    8501 = 47*180 + 41; hence not 47 divides 8501 by NAT_4:9;
    8501 = 53*160 + 21; hence not 53 divides 8501 by NAT_4:9;
    8501 = 59*144 + 5; hence not 59 divides 8501 by NAT_4:9;
    8501 = 61*139 + 22; hence not 61 divides 8501 by NAT_4:9;
    8501 = 67*126 + 59; hence not 67 divides 8501 by NAT_4:9;
    8501 = 71*119 + 52; hence not 71 divides 8501 by NAT_4:9;
    8501 = 73*116 + 33; hence not 73 divides 8501 by NAT_4:9;
    8501 = 79*107 + 48; hence not 79 divides 8501 by NAT_4:9;
    8501 = 83*102 + 35; hence not 83 divides 8501 by NAT_4:9;
    8501 = 89*95 + 46; hence not 89 divides 8501 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8501 & n is prime
  holds not n divides 8501 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
