now :: thesis: ( not 2 divides 569 & not 3 divides 569 & not 5 divides 569 & not 7 divides 569 & not 11 divides 569 & not 13 divides 569 & not 17 divides 569 & not 19 divides 569 & not 23 divides 569 )

569 = (2 * 284) + 1 ;
hence not 2 divides 569 by NAT_4:9; :: thesis: ( not 3 divides 569 & not 5 divides 569 & not 7 divides 569 & not 11 divides 569 & not 13 divides 569 & not 17 divides 569 & not 19 divides 569 & not 23 divides 569 )
569 = (3 * 189) + 2 ;
hence not 3 divides 569 by NAT_4:9; :: thesis: ( not 5 divides 569 & not 7 divides 569 & not 11 divides 569 & not 13 divides 569 & not 17 divides 569 & not 19 divides 569 & not 23 divides 569 )
569 = (5 * 113) + 4 ;
hence not 5 divides 569 by NAT_4:9; :: thesis: ( not 7 divides 569 & not 11 divides 569 & not 13 divides 569 & not 17 divides 569 & not 19 divides 569 & not 23 divides 569 )
569 = (7 * 81) + 2 ;
hence not 7 divides 569 by NAT_4:9; :: thesis: ( not 11 divides 569 & not 13 divides 569 & not 17 divides 569 & not 19 divides 569 & not 23 divides 569 )
569 = (11 * 51) + 8 ;
hence not 11 divides 569 by NAT_4:9; :: thesis: ( not 13 divides 569 & not 17 divides 569 & not 19 divides 569 & not 23 divides 569 )
569 = (13 * 43) + 10 ;
hence not 13 divides 569 by NAT_4:9; :: thesis: ( not 17 divides 569 & not 19 divides 569 & not 23 divides 569 )
569 = (17 * 33) + 8 ;
hence not 17 divides 569 by NAT_4:9; :: thesis: ( not 19 divides 569 & not 23 divides 569 )
569 = (19 * 29) + 18 ;
hence not 19 divides 569 by NAT_4:9; :: thesis: not 23 divides 569
569 = (23 * 24) + 17 ;
hence not 23 divides 569 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 569 & n is prime holds
not n divides 569 by XPRIMET1:18;
hence 569 is prime by NAT_4:14; :: thesis: verum