now :: thesis: ( not 2 divides 829 & not 3 divides 829 & not 5 divides 829 & not 7 divides 829 & not 11 divides 829 & not 13 divides 829 & not 17 divides 829 & not 19 divides 829 & not 23 divides 829 )

829 = (2 * 414) + 1 ;
hence not 2 divides 829 by NAT_4:9; :: thesis: ( not 3 divides 829 & not 5 divides 829 & not 7 divides 829 & not 11 divides 829 & not 13 divides 829 & not 17 divides 829 & not 19 divides 829 & not 23 divides 829 )
829 = (3 * 276) + 1 ;
hence not 3 divides 829 by NAT_4:9; :: thesis: ( not 5 divides 829 & not 7 divides 829 & not 11 divides 829 & not 13 divides 829 & not 17 divides 829 & not 19 divides 829 & not 23 divides 829 )
829 = (5 * 165) + 4 ;
hence not 5 divides 829 by NAT_4:9; :: thesis: ( not 7 divides 829 & not 11 divides 829 & not 13 divides 829 & not 17 divides 829 & not 19 divides 829 & not 23 divides 829 )
829 = (7 * 118) + 3 ;
hence not 7 divides 829 by NAT_4:9; :: thesis: ( not 11 divides 829 & not 13 divides 829 & not 17 divides 829 & not 19 divides 829 & not 23 divides 829 )
829 = (11 * 75) + 4 ;
hence not 11 divides 829 by NAT_4:9; :: thesis: ( not 13 divides 829 & not 17 divides 829 & not 19 divides 829 & not 23 divides 829 )
829 = (13 * 63) + 10 ;
hence not 13 divides 829 by NAT_4:9; :: thesis: ( not 17 divides 829 & not 19 divides 829 & not 23 divides 829 )
829 = (17 * 48) + 13 ;
hence not 17 divides 829 by NAT_4:9; :: thesis: ( not 19 divides 829 & not 23 divides 829 )
829 = (19 * 43) + 12 ;
hence not 19 divides 829 by NAT_4:9; :: thesis: not 23 divides 829
829 = (23 * 36) + 1 ;
hence not 23 divides 829 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 829 & n is prime holds
not n divides 829 by XPRIMET1:18;
hence 829 is prime by NAT_4:14; :: thesis: verum