
theorem
  8543 is prime
proof
  now
    8543 = 2*4271 + 1; hence not 2 divides 8543 by NAT_4:9;
    8543 = 3*2847 + 2; hence not 3 divides 8543 by NAT_4:9;
    8543 = 5*1708 + 3; hence not 5 divides 8543 by NAT_4:9;
    8543 = 7*1220 + 3; hence not 7 divides 8543 by NAT_4:9;
    8543 = 11*776 + 7; hence not 11 divides 8543 by NAT_4:9;
    8543 = 13*657 + 2; hence not 13 divides 8543 by NAT_4:9;
    8543 = 17*502 + 9; hence not 17 divides 8543 by NAT_4:9;
    8543 = 19*449 + 12; hence not 19 divides 8543 by NAT_4:9;
    8543 = 23*371 + 10; hence not 23 divides 8543 by NAT_4:9;
    8543 = 29*294 + 17; hence not 29 divides 8543 by NAT_4:9;
    8543 = 31*275 + 18; hence not 31 divides 8543 by NAT_4:9;
    8543 = 37*230 + 33; hence not 37 divides 8543 by NAT_4:9;
    8543 = 41*208 + 15; hence not 41 divides 8543 by NAT_4:9;
    8543 = 43*198 + 29; hence not 43 divides 8543 by NAT_4:9;
    8543 = 47*181 + 36; hence not 47 divides 8543 by NAT_4:9;
    8543 = 53*161 + 10; hence not 53 divides 8543 by NAT_4:9;
    8543 = 59*144 + 47; hence not 59 divides 8543 by NAT_4:9;
    8543 = 61*140 + 3; hence not 61 divides 8543 by NAT_4:9;
    8543 = 67*127 + 34; hence not 67 divides 8543 by NAT_4:9;
    8543 = 71*120 + 23; hence not 71 divides 8543 by NAT_4:9;
    8543 = 73*117 + 2; hence not 73 divides 8543 by NAT_4:9;
    8543 = 79*108 + 11; hence not 79 divides 8543 by NAT_4:9;
    8543 = 83*102 + 77; hence not 83 divides 8543 by NAT_4:9;
    8543 = 89*95 + 88; hence not 89 divides 8543 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8543 & n is prime
  holds not n divides 8543 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
