now :: thesis: ( not 2 divides 571 & not 3 divides 571 & not 5 divides 571 & not 7 divides 571 & not 11 divides 571 & not 13 divides 571 & not 17 divides 571 & not 19 divides 571 & not 23 divides 571 )

571 = (2 * 285) + 1 ;
hence not 2 divides 571 by NAT_4:9; :: thesis: ( not 3 divides 571 & not 5 divides 571 & not 7 divides 571 & not 11 divides 571 & not 13 divides 571 & not 17 divides 571 & not 19 divides 571 & not 23 divides 571 )
571 = (3 * 190) + 1 ;
hence not 3 divides 571 by NAT_4:9; :: thesis: ( not 5 divides 571 & not 7 divides 571 & not 11 divides 571 & not 13 divides 571 & not 17 divides 571 & not 19 divides 571 & not 23 divides 571 )
571 = (5 * 114) + 1 ;
hence not 5 divides 571 by NAT_4:9; :: thesis: ( not 7 divides 571 & not 11 divides 571 & not 13 divides 571 & not 17 divides 571 & not 19 divides 571 & not 23 divides 571 )
571 = (7 * 81) + 4 ;
hence not 7 divides 571 by NAT_4:9; :: thesis: ( not 11 divides 571 & not 13 divides 571 & not 17 divides 571 & not 19 divides 571 & not 23 divides 571 )
571 = (11 * 51) + 10 ;
hence not 11 divides 571 by NAT_4:9; :: thesis: ( not 13 divides 571 & not 17 divides 571 & not 19 divides 571 & not 23 divides 571 )
571 = (13 * 43) + 12 ;
hence not 13 divides 571 by NAT_4:9; :: thesis: ( not 17 divides 571 & not 19 divides 571 & not 23 divides 571 )
571 = (17 * 33) + 10 ;
hence not 17 divides 571 by NAT_4:9; :: thesis: ( not 19 divides 571 & not 23 divides 571 )
571 = (19 * 30) + 1 ;
hence not 19 divides 571 by NAT_4:9; :: thesis: not 23 divides 571
571 = (23 * 24) + 19 ;
hence not 23 divides 571 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 571 & n is prime holds
not n divides 571 by XPRIMET1:18;
hence 571 is prime by NAT_4:14; :: thesis: verum