
theorem
  8831 is prime
proof
  now
    8831 = 2*4415 + 1; hence not 2 divides 8831 by NAT_4:9;
    8831 = 3*2943 + 2; hence not 3 divides 8831 by NAT_4:9;
    8831 = 5*1766 + 1; hence not 5 divides 8831 by NAT_4:9;
    8831 = 7*1261 + 4; hence not 7 divides 8831 by NAT_4:9;
    8831 = 11*802 + 9; hence not 11 divides 8831 by NAT_4:9;
    8831 = 13*679 + 4; hence not 13 divides 8831 by NAT_4:9;
    8831 = 17*519 + 8; hence not 17 divides 8831 by NAT_4:9;
    8831 = 19*464 + 15; hence not 19 divides 8831 by NAT_4:9;
    8831 = 23*383 + 22; hence not 23 divides 8831 by NAT_4:9;
    8831 = 29*304 + 15; hence not 29 divides 8831 by NAT_4:9;
    8831 = 31*284 + 27; hence not 31 divides 8831 by NAT_4:9;
    8831 = 37*238 + 25; hence not 37 divides 8831 by NAT_4:9;
    8831 = 41*215 + 16; hence not 41 divides 8831 by NAT_4:9;
    8831 = 43*205 + 16; hence not 43 divides 8831 by NAT_4:9;
    8831 = 47*187 + 42; hence not 47 divides 8831 by NAT_4:9;
    8831 = 53*166 + 33; hence not 53 divides 8831 by NAT_4:9;
    8831 = 59*149 + 40; hence not 59 divides 8831 by NAT_4:9;
    8831 = 61*144 + 47; hence not 61 divides 8831 by NAT_4:9;
    8831 = 67*131 + 54; hence not 67 divides 8831 by NAT_4:9;
    8831 = 71*124 + 27; hence not 71 divides 8831 by NAT_4:9;
    8831 = 73*120 + 71; hence not 73 divides 8831 by NAT_4:9;
    8831 = 79*111 + 62; hence not 79 divides 8831 by NAT_4:9;
    8831 = 83*106 + 33; hence not 83 divides 8831 by NAT_4:9;
    8831 = 89*99 + 20; hence not 89 divides 8831 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8831 & n is prime
  holds not n divides 8831 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
