
theorem
  557 is prime
proof
  now
    557 = 2*278 + 1; hence not 2 divides 557 by NAT_4:9;
    557 = 3*185 + 2; hence not 3 divides 557 by NAT_4:9;
    557 = 5*111 + 2; hence not 5 divides 557 by NAT_4:9;
    557 = 7*79 + 4; hence not 7 divides 557 by NAT_4:9;
    557 = 11*50 + 7; hence not 11 divides 557 by NAT_4:9;
    557 = 13*42 + 11; hence not 13 divides 557 by NAT_4:9;
    557 = 17*32 + 13; hence not 17 divides 557 by NAT_4:9;
    557 = 19*29 + 6; hence not 19 divides 557 by NAT_4:9;
    557 = 23*24 + 5; hence not 23 divides 557 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 557 & n is prime
  holds not n divides 557 by XPRIMET1:18;
  hence thesis by NAT_4:14;
