now :: thesis: ( not 2 divides 809 & not 3 divides 809 & not 5 divides 809 & not 7 divides 809 & not 11 divides 809 & not 13 divides 809 & not 17 divides 809 & not 19 divides 809 & not 23 divides 809 )

809 = (2 * 404) + 1 ;
hence not 2 divides 809 by NAT_4:9; :: thesis: ( not 3 divides 809 & not 5 divides 809 & not 7 divides 809 & not 11 divides 809 & not 13 divides 809 & not 17 divides 809 & not 19 divides 809 & not 23 divides 809 )
809 = (3 * 269) + 2 ;
hence not 3 divides 809 by NAT_4:9; :: thesis: ( not 5 divides 809 & not 7 divides 809 & not 11 divides 809 & not 13 divides 809 & not 17 divides 809 & not 19 divides 809 & not 23 divides 809 )
809 = (5 * 161) + 4 ;
hence not 5 divides 809 by NAT_4:9; :: thesis: ( not 7 divides 809 & not 11 divides 809 & not 13 divides 809 & not 17 divides 809 & not 19 divides 809 & not 23 divides 809 )
809 = (7 * 115) + 4 ;
hence not 7 divides 809 by NAT_4:9; :: thesis: ( not 11 divides 809 & not 13 divides 809 & not 17 divides 809 & not 19 divides 809 & not 23 divides 809 )
809 = (11 * 73) + 6 ;
hence not 11 divides 809 by NAT_4:9; :: thesis: ( not 13 divides 809 & not 17 divides 809 & not 19 divides 809 & not 23 divides 809 )
809 = (13 * 62) + 3 ;
hence not 13 divides 809 by NAT_4:9; :: thesis: ( not 17 divides 809 & not 19 divides 809 & not 23 divides 809 )
809 = (17 * 47) + 10 ;
hence not 17 divides 809 by NAT_4:9; :: thesis: ( not 19 divides 809 & not 23 divides 809 )
809 = (19 * 42) + 11 ;
hence not 19 divides 809 by NAT_4:9; :: thesis: not 23 divides 809
809 = (23 * 35) + 4 ;
hence not 23 divides 809 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 809 & n is prime holds
not n divides 809 by XPRIMET1:18;
hence 809 is prime by NAT_4:14; :: thesis: verum