now :: thesis: ( not 2 divides 797 & not 3 divides 797 & not 5 divides 797 & not 7 divides 797 & not 11 divides 797 & not 13 divides 797 & not 17 divides 797 & not 19 divides 797 & not 23 divides 797 )

797 = (2 * 398) + 1 ;
hence not 2 divides 797 by NAT_4:9; :: thesis: ( not 3 divides 797 & not 5 divides 797 & not 7 divides 797 & not 11 divides 797 & not 13 divides 797 & not 17 divides 797 & not 19 divides 797 & not 23 divides 797 )
797 = (3 * 265) + 2 ;
hence not 3 divides 797 by NAT_4:9; :: thesis: ( not 5 divides 797 & not 7 divides 797 & not 11 divides 797 & not 13 divides 797 & not 17 divides 797 & not 19 divides 797 & not 23 divides 797 )
797 = (5 * 159) + 2 ;
hence not 5 divides 797 by NAT_4:9; :: thesis: ( not 7 divides 797 & not 11 divides 797 & not 13 divides 797 & not 17 divides 797 & not 19 divides 797 & not 23 divides 797 )
797 = (7 * 113) + 6 ;
hence not 7 divides 797 by NAT_4:9; :: thesis: ( not 11 divides 797 & not 13 divides 797 & not 17 divides 797 & not 19 divides 797 & not 23 divides 797 )
797 = (11 * 72) + 5 ;
hence not 11 divides 797 by NAT_4:9; :: thesis: ( not 13 divides 797 & not 17 divides 797 & not 19 divides 797 & not 23 divides 797 )
797 = (13 * 61) + 4 ;
hence not 13 divides 797 by NAT_4:9; :: thesis: ( not 17 divides 797 & not 19 divides 797 & not 23 divides 797 )
797 = (17 * 46) + 15 ;
hence not 17 divides 797 by NAT_4:9; :: thesis: ( not 19 divides 797 & not 23 divides 797 )
797 = (19 * 41) + 18 ;
hence not 19 divides 797 by NAT_4:9; :: thesis: not 23 divides 797
797 = (23 * 34) + 15 ;
hence not 23 divides 797 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 797 & n is prime holds
not n divides 797 by XPRIMET1:18;
hence 797 is prime by NAT_4:14; :: thesis: verum