now :: thesis: ( not 2 divides 509 & not 3 divides 509 & not 5 divides 509 & not 7 divides 509 & not 11 divides 509 & not 13 divides 509 & not 17 divides 509 & not 19 divides 509 )

509 = (2 * 254) + 1 ;
hence not 2 divides 509 by NAT_4:9; :: thesis: ( not 3 divides 509 & not 5 divides 509 & not 7 divides 509 & not 11 divides 509 & not 13 divides 509 & not 17 divides 509 & not 19 divides 509 )
509 = (3 * 169) + 2 ;
hence not 3 divides 509 by NAT_4:9; :: thesis: ( not 5 divides 509 & not 7 divides 509 & not 11 divides 509 & not 13 divides 509 & not 17 divides 509 & not 19 divides 509 )
509 = (5 * 101) + 4 ;
hence not 5 divides 509 by NAT_4:9; :: thesis: ( not 7 divides 509 & not 11 divides 509 & not 13 divides 509 & not 17 divides 509 & not 19 divides 509 )
509 = (7 * 72) + 5 ;
hence not 7 divides 509 by NAT_4:9; :: thesis: ( not 11 divides 509 & not 13 divides 509 & not 17 divides 509 & not 19 divides 509 )
509 = (11 * 46) + 3 ;
hence not 11 divides 509 by NAT_4:9; :: thesis: ( not 13 divides 509 & not 17 divides 509 & not 19 divides 509 )
509 = (13 * 39) + 2 ;
hence not 13 divides 509 by NAT_4:9; :: thesis: ( not 17 divides 509 & not 19 divides 509 )
509 = (17 * 29) + 16 ;
hence not 17 divides 509 by NAT_4:9; :: thesis: not 19 divides 509
509 = (19 * 26) + 15 ;
hence not 19 divides 509 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 509 & n is prime holds
not n divides 509 by XPRIMET1:16;
hence 509 is prime by NAT_4:14; :: thesis: verum