
theorem
  683 is prime
proof
  now
    683 = 2*341 + 1; hence not 2 divides 683 by NAT_4:9;
    683 = 3*227 + 2; hence not 3 divides 683 by NAT_4:9;
    683 = 5*136 + 3; hence not 5 divides 683 by NAT_4:9;
    683 = 7*97 + 4; hence not 7 divides 683 by NAT_4:9;
    683 = 11*62 + 1; hence not 11 divides 683 by NAT_4:9;
    683 = 13*52 + 7; hence not 13 divides 683 by NAT_4:9;
    683 = 17*40 + 3; hence not 17 divides 683 by NAT_4:9;
    683 = 19*35 + 18; hence not 19 divides 683 by NAT_4:9;
    683 = 23*29 + 16; hence not 23 divides 683 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 683 & n is prime
  holds not n divides 683 by XPRIMET1:18;
  hence thesis by NAT_4:14;
end;
