
theorem
  4177 is prime
proof
  now
    4177 = 2*2088 + 1; hence not 2 divides 4177 by NAT_4:9;
    4177 = 3*1392 + 1; hence not 3 divides 4177 by NAT_4:9;
    4177 = 5*835 + 2; hence not 5 divides 4177 by NAT_4:9;
    4177 = 7*596 + 5; hence not 7 divides 4177 by NAT_4:9;
    4177 = 11*379 + 8; hence not 11 divides 4177 by NAT_4:9;
    4177 = 13*321 + 4; hence not 13 divides 4177 by NAT_4:9;
    4177 = 17*245 + 12; hence not 17 divides 4177 by NAT_4:9;
    4177 = 19*219 + 16; hence not 19 divides 4177 by NAT_4:9;
    4177 = 23*181 + 14; hence not 23 divides 4177 by NAT_4:9;
    4177 = 29*144 + 1; hence not 29 divides 4177 by NAT_4:9;
    4177 = 31*134 + 23; hence not 31 divides 4177 by NAT_4:9;
    4177 = 37*112 + 33; hence not 37 divides 4177 by NAT_4:9;
    4177 = 41*101 + 36; hence not 41 divides 4177 by NAT_4:9;
    4177 = 43*97 + 6; hence not 43 divides 4177 by NAT_4:9;
    4177 = 47*88 + 41; hence not 47 divides 4177 by NAT_4:9;
    4177 = 53*78 + 43; hence not 53 divides 4177 by NAT_4:9;
    4177 = 59*70 + 47; hence not 59 divides 4177 by NAT_4:9;
    4177 = 61*68 + 29; hence not 61 divides 4177 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 4177 & n is prime
  holds not n divides 4177 by XPRIMET1:36;
  hence thesis by NAT_4:14;
end;
