
theorem
  8821 is prime
proof
  now
    8821 = 2*4410 + 1; hence not 2 divides 8821 by NAT_4:9;
    8821 = 3*2940 + 1; hence not 3 divides 8821 by NAT_4:9;
    8821 = 5*1764 + 1; hence not 5 divides 8821 by NAT_4:9;
    8821 = 7*1260 + 1; hence not 7 divides 8821 by NAT_4:9;
    8821 = 11*801 + 10; hence not 11 divides 8821 by NAT_4:9;
    8821 = 13*678 + 7; hence not 13 divides 8821 by NAT_4:9;
    8821 = 17*518 + 15; hence not 17 divides 8821 by NAT_4:9;
    8821 = 19*464 + 5; hence not 19 divides 8821 by NAT_4:9;
    8821 = 23*383 + 12; hence not 23 divides 8821 by NAT_4:9;
    8821 = 29*304 + 5; hence not 29 divides 8821 by NAT_4:9;
    8821 = 31*284 + 17; hence not 31 divides 8821 by NAT_4:9;
    8821 = 37*238 + 15; hence not 37 divides 8821 by NAT_4:9;
    8821 = 41*215 + 6; hence not 41 divides 8821 by NAT_4:9;
    8821 = 43*205 + 6; hence not 43 divides 8821 by NAT_4:9;
    8821 = 47*187 + 32; hence not 47 divides 8821 by NAT_4:9;
    8821 = 53*166 + 23; hence not 53 divides 8821 by NAT_4:9;
    8821 = 59*149 + 30; hence not 59 divides 8821 by NAT_4:9;
    8821 = 61*144 + 37; hence not 61 divides 8821 by NAT_4:9;
    8821 = 67*131 + 44; hence not 67 divides 8821 by NAT_4:9;
    8821 = 71*124 + 17; hence not 71 divides 8821 by NAT_4:9;
    8821 = 73*120 + 61; hence not 73 divides 8821 by NAT_4:9;
    8821 = 79*111 + 52; hence not 79 divides 8821 by NAT_4:9;
    8821 = 83*106 + 23; hence not 83 divides 8821 by NAT_4:9;
    8821 = 89*99 + 10; hence not 89 divides 8821 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 8821 & n is prime
  holds not n divides 8821 by XPRIMET1:48;
  hence thesis by NAT_4:14;
end;
