now :: thesis: ( not 2 divides 521 & not 3 divides 521 & not 5 divides 521 & not 7 divides 521 & not 11 divides 521 & not 13 divides 521 & not 17 divides 521 & not 19 divides 521 )

521 = (2 * 260) + 1 ;
hence not 2 divides 521 by NAT_4:9; :: thesis: ( not 3 divides 521 & not 5 divides 521 & not 7 divides 521 & not 11 divides 521 & not 13 divides 521 & not 17 divides 521 & not 19 divides 521 )
521 = (3 * 173) + 2 ;
hence not 3 divides 521 by NAT_4:9; :: thesis: ( not 5 divides 521 & not 7 divides 521 & not 11 divides 521 & not 13 divides 521 & not 17 divides 521 & not 19 divides 521 )
521 = (5 * 104) + 1 ;
hence not 5 divides 521 by NAT_4:9; :: thesis: ( not 7 divides 521 & not 11 divides 521 & not 13 divides 521 & not 17 divides 521 & not 19 divides 521 )
521 = (7 * 74) + 3 ;
hence not 7 divides 521 by NAT_4:9; :: thesis: ( not 11 divides 521 & not 13 divides 521 & not 17 divides 521 & not 19 divides 521 )
521 = (11 * 47) + 4 ;
hence not 11 divides 521 by NAT_4:9; :: thesis: ( not 13 divides 521 & not 17 divides 521 & not 19 divides 521 )
521 = (13 * 40) + 1 ;
hence not 13 divides 521 by NAT_4:9; :: thesis: ( not 17 divides 521 & not 19 divides 521 )
521 = (17 * 30) + 11 ;
hence not 17 divides 521 by NAT_4:9; :: thesis: not 19 divides 521
521 = (19 * 27) + 8 ;
hence not 19 divides 521 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 521 & n is prime holds
not n divides 521 by XPRIMET1:16;
hence 521 is prime by NAT_4:14; :: thesis: verum