
theorem
  8353 is prime
proof
  now
    8353 = 2*4176 + 1; hence not 2 divides 8353 by NAT_4:9;
    8353 = 3*2784 + 1; hence not 3 divides 8353 by NAT_4:9;
    8353 = 5*1670 + 3; hence not 5 divides 8353 by NAT_4:9;
    8353 = 7*1193 + 2; hence not 7 divides 8353 by NAT_4:9;
    8353 = 11*759 + 4; hence not 11 divides 8353 by NAT_4:9;
    8353 = 13*642 + 7; hence not 13 divides 8353 by NAT_4:9;
    8353 = 17*491 + 6; hence not 17 divides 8353 by NAT_4:9;
    8353 = 19*439 + 12; hence not 19 divides 8353 by NAT_4:9;
    8353 = 23*363 + 4; hence not 23 divides 8353 by NAT_4:9;
    8353 = 29*288 + 1; hence not 29 divides 8353 by NAT_4:9;
    8353 = 31*269 + 14; hence not 31 divides 8353 by NAT_4:9;
    8353 = 37*225 + 28; hence not 37 divides 8353 by NAT_4:9;
    8353 = 41*203 + 30; hence not 41 divides 8353 by NAT_4:9;
    8353 = 43*194 + 11; hence not 43 divides 8353 by NAT_4:9;
    8353 = 47*177 + 34; hence not 47 divides 8353 by NAT_4:9;
    8353 = 53*157 + 32; hence not 53 divides 8353 by NAT_4:9;
    8353 = 59*141 + 34; hence not 59 divides 8353 by NAT_4:9;
    8353 = 61*136 + 57; hence not 61 divides 8353 by NAT_4:9;
    8353 = 67*124 + 45; hence not 67 divides 8353 by NAT_4:9;
    8353 = 71*117 + 46; hence not 71 divides 8353 by NAT_4:9;
    8353 = 73*114 + 31; hence not 73 divides 8353 by NAT_4:9;
    8353 = 79*105 + 58; hence not 79 divides 8353 by NAT_4:9;
    8353 = 83*100 + 53; hence not 83 divides 8353 by NAT_4:9;
    8353 = 89*93 + 76; hence not 89 divides 8353 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8353 & n is prime
  holds not n divides 8353 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
