
theorem
  8263 is prime
proof
  now
    8263 = 2*4131 + 1; hence not 2 divides 8263 by NAT_4:9;
    8263 = 3*2754 + 1; hence not 3 divides 8263 by NAT_4:9;
    8263 = 5*1652 + 3; hence not 5 divides 8263 by NAT_4:9;
    8263 = 7*1180 + 3; hence not 7 divides 8263 by NAT_4:9;
    8263 = 11*751 + 2; hence not 11 divides 8263 by NAT_4:9;
    8263 = 13*635 + 8; hence not 13 divides 8263 by NAT_4:9;
    8263 = 17*486 + 1; hence not 17 divides 8263 by NAT_4:9;
    8263 = 19*434 + 17; hence not 19 divides 8263 by NAT_4:9;
    8263 = 23*359 + 6; hence not 23 divides 8263 by NAT_4:9;
    8263 = 29*284 + 27; hence not 29 divides 8263 by NAT_4:9;
    8263 = 31*266 + 17; hence not 31 divides 8263 by NAT_4:9;
    8263 = 37*223 + 12; hence not 37 divides 8263 by NAT_4:9;
    8263 = 41*201 + 22; hence not 41 divides 8263 by NAT_4:9;
    8263 = 43*192 + 7; hence not 43 divides 8263 by NAT_4:9;
    8263 = 47*175 + 38; hence not 47 divides 8263 by NAT_4:9;
    8263 = 53*155 + 48; hence not 53 divides 8263 by NAT_4:9;
    8263 = 59*140 + 3; hence not 59 divides 8263 by NAT_4:9;
    8263 = 61*135 + 28; hence not 61 divides 8263 by NAT_4:9;
    8263 = 67*123 + 22; hence not 67 divides 8263 by NAT_4:9;
    8263 = 71*116 + 27; hence not 71 divides 8263 by NAT_4:9;
    8263 = 73*113 + 14; hence not 73 divides 8263 by NAT_4:9;
    8263 = 79*104 + 47; hence not 79 divides 8263 by NAT_4:9;
    8263 = 83*99 + 46; hence not 83 divides 8263 by NAT_4:9;
    8263 = 89*92 + 75; hence not 89 divides 8263 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8263 & n is prime
  holds not n divides 8263 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
