
theorem
  5581 is prime
proof
  now
    5581 = 2*2790 + 1; hence not 2 divides 5581 by NAT_4:9;
    5581 = 3*1860 + 1; hence not 3 divides 5581 by NAT_4:9;
    5581 = 5*1116 + 1; hence not 5 divides 5581 by NAT_4:9;
    5581 = 7*797 + 2; hence not 7 divides 5581 by NAT_4:9;
    5581 = 11*507 + 4; hence not 11 divides 5581 by NAT_4:9;
    5581 = 13*429 + 4; hence not 13 divides 5581 by NAT_4:9;
    5581 = 17*328 + 5; hence not 17 divides 5581 by NAT_4:9;
    5581 = 19*293 + 14; hence not 19 divides 5581 by NAT_4:9;
    5581 = 23*242 + 15; hence not 23 divides 5581 by NAT_4:9;
    5581 = 29*192 + 13; hence not 29 divides 5581 by NAT_4:9;
    5581 = 31*180 + 1; hence not 31 divides 5581 by NAT_4:9;
    5581 = 37*150 + 31; hence not 37 divides 5581 by NAT_4:9;
    5581 = 41*136 + 5; hence not 41 divides 5581 by NAT_4:9;
    5581 = 43*129 + 34; hence not 43 divides 5581 by NAT_4:9;
    5581 = 47*118 + 35; hence not 47 divides 5581 by NAT_4:9;
    5581 = 53*105 + 16; hence not 53 divides 5581 by NAT_4:9;
    5581 = 59*94 + 35; hence not 59 divides 5581 by NAT_4:9;
    5581 = 61*91 + 30; hence not 61 divides 5581 by NAT_4:9;
    5581 = 67*83 + 20; hence not 67 divides 5581 by NAT_4:9;
    5581 = 71*78 + 43; hence not 71 divides 5581 by NAT_4:9;
    5581 = 73*76 + 33; hence not 73 divides 5581 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 5581 & n is prime
  holds not n divides 5581 by XPRIMET1:42;
  hence thesis by NAT_4:14;
end;
