now :: thesis: ( not 2 divides 769 & not 3 divides 769 & not 5 divides 769 & not 7 divides 769 & not 11 divides 769 & not 13 divides 769 & not 17 divides 769 & not 19 divides 769 & not 23 divides 769 )

769 = (2 * 384) + 1 ;
hence not 2 divides 769 by NAT_4:9; :: thesis: ( not 3 divides 769 & not 5 divides 769 & not 7 divides 769 & not 11 divides 769 & not 13 divides 769 & not 17 divides 769 & not 19 divides 769 & not 23 divides 769 )
769 = (3 * 256) + 1 ;
hence not 3 divides 769 by NAT_4:9; :: thesis: ( not 5 divides 769 & not 7 divides 769 & not 11 divides 769 & not 13 divides 769 & not 17 divides 769 & not 19 divides 769 & not 23 divides 769 )
769 = (5 * 153) + 4 ;
hence not 5 divides 769 by NAT_4:9; :: thesis: ( not 7 divides 769 & not 11 divides 769 & not 13 divides 769 & not 17 divides 769 & not 19 divides 769 & not 23 divides 769 )
769 = (7 * 109) + 6 ;
hence not 7 divides 769 by NAT_4:9; :: thesis: ( not 11 divides 769 & not 13 divides 769 & not 17 divides 769 & not 19 divides 769 & not 23 divides 769 )
769 = (11 * 69) + 10 ;
hence not 11 divides 769 by NAT_4:9; :: thesis: ( not 13 divides 769 & not 17 divides 769 & not 19 divides 769 & not 23 divides 769 )
769 = (13 * 59) + 2 ;
hence not 13 divides 769 by NAT_4:9; :: thesis: ( not 17 divides 769 & not 19 divides 769 & not 23 divides 769 )
769 = (17 * 45) + 4 ;
hence not 17 divides 769 by NAT_4:9; :: thesis: ( not 19 divides 769 & not 23 divides 769 )
769 = (19 * 40) + 9 ;
hence not 19 divides 769 by NAT_4:9; :: thesis: not 23 divides 769
769 = (23 * 33) + 10 ;
hence not 23 divides 769 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 769 & n is prime holds
not n divides 769 by XPRIMET1:18;
hence 769 is prime by NAT_4:14; :: thesis: verum