
theorem
  2557 is prime
proof
  now
    2557 = 2*1278 + 1; hence not 2 divides 2557 by NAT_4:9;
    2557 = 3*852 + 1; hence not 3 divides 2557 by NAT_4:9;
    2557 = 5*511 + 2; hence not 5 divides 2557 by NAT_4:9;
    2557 = 7*365 + 2; hence not 7 divides 2557 by NAT_4:9;
    2557 = 11*232 + 5; hence not 11 divides 2557 by NAT_4:9;
    2557 = 13*196 + 9; hence not 13 divides 2557 by NAT_4:9;
    2557 = 17*150 + 7; hence not 17 divides 2557 by NAT_4:9;
    2557 = 19*134 + 11; hence not 19 divides 2557 by NAT_4:9;
    2557 = 23*111 + 4; hence not 23 divides 2557 by NAT_4:9;
    2557 = 29*88 + 5; hence not 29 divides 2557 by NAT_4:9;
    2557 = 31*82 + 15; hence not 31 divides 2557 by NAT_4:9;
    2557 = 37*69 + 4; hence not 37 divides 2557 by NAT_4:9;
    2557 = 41*62 + 15; hence not 41 divides 2557 by NAT_4:9;
    2557 = 43*59 + 20; hence not 43 divides 2557 by NAT_4:9;
    2557 = 47*54 + 19; hence not 47 divides 2557 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 2557 & n is prime
  holds not n divides 2557 by XPRIMET1:30;
  hence thesis by NAT_4:14;
end;
