now :: thesis: ( not 2 divides 641 & not 3 divides 641 & not 5 divides 641 & not 7 divides 641 & not 11 divides 641 & not 13 divides 641 & not 17 divides 641 & not 19 divides 641 & not 23 divides 641 )

641 = (2 * 320) + 1 ;
hence not 2 divides 641 by NAT_4:9; :: thesis: ( not 3 divides 641 & not 5 divides 641 & not 7 divides 641 & not 11 divides 641 & not 13 divides 641 & not 17 divides 641 & not 19 divides 641 & not 23 divides 641 )
641 = (3 * 213) + 2 ;
hence not 3 divides 641 by NAT_4:9; :: thesis: ( not 5 divides 641 & not 7 divides 641 & not 11 divides 641 & not 13 divides 641 & not 17 divides 641 & not 19 divides 641 & not 23 divides 641 )
641 = (5 * 128) + 1 ;
hence not 5 divides 641 by NAT_4:9; :: thesis: ( not 7 divides 641 & not 11 divides 641 & not 13 divides 641 & not 17 divides 641 & not 19 divides 641 & not 23 divides 641 )
641 = (7 * 91) + 4 ;
hence not 7 divides 641 by NAT_4:9; :: thesis: ( not 11 divides 641 & not 13 divides 641 & not 17 divides 641 & not 19 divides 641 & not 23 divides 641 )
641 = (11 * 58) + 3 ;
hence not 11 divides 641 by NAT_4:9; :: thesis: ( not 13 divides 641 & not 17 divides 641 & not 19 divides 641 & not 23 divides 641 )
641 = (13 * 49) + 4 ;
hence not 13 divides 641 by NAT_4:9; :: thesis: ( not 17 divides 641 & not 19 divides 641 & not 23 divides 641 )
641 = (17 * 37) + 12 ;
hence not 17 divides 641 by NAT_4:9; :: thesis: ( not 19 divides 641 & not 23 divides 641 )
641 = (19 * 33) + 14 ;
hence not 19 divides 641 by NAT_4:9; :: thesis: not 23 divides 641
641 = (23 * 27) + 20 ;
hence not 23 divides 641 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 641 & n is prime holds
not n divides 641 by XPRIMET1:18;
hence 641 is prime by NAT_4:14; :: thesis: verum