now :: thesis: ( not 2 divides 647 & not 3 divides 647 & not 5 divides 647 & not 7 divides 647 & not 11 divides 647 & not 13 divides 647 & not 17 divides 647 & not 19 divides 647 & not 23 divides 647 )

647 = (2 * 323) + 1 ;
hence not 2 divides 647 by NAT_4:9; :: thesis: ( not 3 divides 647 & not 5 divides 647 & not 7 divides 647 & not 11 divides 647 & not 13 divides 647 & not 17 divides 647 & not 19 divides 647 & not 23 divides 647 )
647 = (3 * 215) + 2 ;
hence not 3 divides 647 by NAT_4:9; :: thesis: ( not 5 divides 647 & not 7 divides 647 & not 11 divides 647 & not 13 divides 647 & not 17 divides 647 & not 19 divides 647 & not 23 divides 647 )
647 = (5 * 129) + 2 ;
hence not 5 divides 647 by NAT_4:9; :: thesis: ( not 7 divides 647 & not 11 divides 647 & not 13 divides 647 & not 17 divides 647 & not 19 divides 647 & not 23 divides 647 )
647 = (7 * 92) + 3 ;
hence not 7 divides 647 by NAT_4:9; :: thesis: ( not 11 divides 647 & not 13 divides 647 & not 17 divides 647 & not 19 divides 647 & not 23 divides 647 )
647 = (11 * 58) + 9 ;
hence not 11 divides 647 by NAT_4:9; :: thesis: ( not 13 divides 647 & not 17 divides 647 & not 19 divides 647 & not 23 divides 647 )
647 = (13 * 49) + 10 ;
hence not 13 divides 647 by NAT_4:9; :: thesis: ( not 17 divides 647 & not 19 divides 647 & not 23 divides 647 )
647 = (17 * 38) + 1 ;
hence not 17 divides 647 by NAT_4:9; :: thesis: ( not 19 divides 647 & not 23 divides 647 )
647 = (19 * 34) + 1 ;
hence not 19 divides 647 by NAT_4:9; :: thesis: not 23 divides 647
647 = (23 * 28) + 3 ;
hence not 23 divides 647 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 647 & n is prime holds
not n divides 647 by XPRIMET1:18;
hence 647 is prime by NAT_4:14; :: thesis: verum