643 = (2 * 321) + 1 ;
hence not 2 divides 643 by NAT_4:9; :: thesis:
643 = (3 * 214) + 1 ;
hence not 3 divides 643 by NAT_4:9; :: thesis:
643 = (5 * 128) + 3 ;
hence not 5 divides 643 by NAT_4:9; :: thesis:
643 = (7 * 91) + 6 ;
hence not 7 divides 643 by NAT_4:9; :: thesis:
643 = (11 * 58) + 5 ;
hence not 11 divides 643 by NAT_4:9; :: thesis:
643 = (13 * 49) + 6 ;
hence not 13 divides 643 by NAT_4:9; :: thesis:
643 = (17 * 37) + 14 ;
hence not 17 divides 643 by NAT_4:9; :: thesis:
643 = (19 * 33) + 16 ;
hence not 19 divides 643 by NAT_4:9; :: thesis:
643 = (23 * 27) + 22 ;
hence not 23 divides 643 by NAT_4:9; :: thesis:

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 643 is prime by NAT_4:14; :: thesis: