
theorem
  677 is prime
proof
  now
    677 = 2*338 + 1; hence not 2 divides 677 by NAT_4:9;
    677 = 3*225 + 2; hence not 3 divides 677 by NAT_4:9;
    677 = 5*135 + 2; hence not 5 divides 677 by NAT_4:9;
    677 = 7*96 + 5; hence not 7 divides 677 by NAT_4:9;
    677 = 11*61 + 6; hence not 11 divides 677 by NAT_4:9;
    677 = 13*52 + 1; hence not 13 divides 677 by NAT_4:9;
    677 = 17*39 + 14; hence not 17 divides 677 by NAT_4:9;
    677 = 19*35 + 12; hence not 19 divides 677 by NAT_4:9;
    677 = 23*29 + 10; hence not 23 divides 677 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 677 & n is prime
  holds not n divides 677 by XPRIMET1:18;
  hence thesis by NAT_4:14;
