
theorem
  643 is prime
proof
  now
    643 = 2*321 + 1; hence not 2 divides 643 by NAT_4:9;
    643 = 3*214 + 1; hence not 3 divides 643 by NAT_4:9;
    643 = 5*128 + 3; hence not 5 divides 643 by NAT_4:9;
    643 = 7*91 + 6; hence not 7 divides 643 by NAT_4:9;
    643 = 11*58 + 5; hence not 11 divides 643 by NAT_4:9;
    643 = 13*49 + 6; hence not 13 divides 643 by NAT_4:9;
    643 = 17*37 + 14; hence not 17 divides 643 by NAT_4:9;
    643 = 19*33 + 16; hence not 19 divides 643 by NAT_4:9;
    643 = 23*27 + 22; hence not 23 divides 643 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 643 & n is prime
  holds not n divides 643 by XPRIMET1:18;
  hence thesis by NAT_4:14;
