now :: thesis: ( not 2 divides 677 & not 3 divides 677 & not 5 divides 677 & not 7 divides 677 & not 11 divides 677 & not 13 divides 677 & not 17 divides 677 & not 19 divides 677 & not 23 divides 677 )

677 = (2 * 338) + 1 ;
hence not 2 divides 677 by NAT_4:9; :: thesis: ( not 3 divides 677 & not 5 divides 677 & not 7 divides 677 & not 11 divides 677 & not 13 divides 677 & not 17 divides 677 & not 19 divides 677 & not 23 divides 677 )
677 = (3 * 225) + 2 ;
hence not 3 divides 677 by NAT_4:9; :: thesis: ( not 5 divides 677 & not 7 divides 677 & not 11 divides 677 & not 13 divides 677 & not 17 divides 677 & not 19 divides 677 & not 23 divides 677 )
677 = (5 * 135) + 2 ;
hence not 5 divides 677 by NAT_4:9; :: thesis: ( not 7 divides 677 & not 11 divides 677 & not 13 divides 677 & not 17 divides 677 & not 19 divides 677 & not 23 divides 677 )
677 = (7 * 96) + 5 ;
hence not 7 divides 677 by NAT_4:9; :: thesis: ( not 11 divides 677 & not 13 divides 677 & not 17 divides 677 & not 19 divides 677 & not 23 divides 677 )
677 = (11 * 61) + 6 ;
hence not 11 divides 677 by NAT_4:9; :: thesis: ( not 13 divides 677 & not 17 divides 677 & not 19 divides 677 & not 23 divides 677 )
677 = (13 * 52) + 1 ;
hence not 13 divides 677 by NAT_4:9; :: thesis: ( not 17 divides 677 & not 19 divides 677 & not 23 divides 677 )
677 = (17 * 39) + 14 ;
hence not 17 divides 677 by NAT_4:9; :: thesis: ( not 19 divides 677 & not 23 divides 677 )
677 = (19 * 35) + 12 ;
hence not 19 divides 677 by NAT_4:9; :: thesis: not 23 divides 677
677 = (23 * 29) + 10 ;
hence not 23 divides 677 by NAT_4:9; :: thesis: verum

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 677 is prime by NAT_4:14; :: thesis: verum