now :: thesis: ( not 2 divides 499 & not 3 divides 499 & not 5 divides 499 & not 7 divides 499 & not 11 divides 499 & not 13 divides 499 & not 17 divides 499 & not 19 divides 499 )

499 = (2 * 249) + 1 ;
hence not 2 divides 499 by NAT_4:9; :: thesis: ( not 3 divides 499 & not 5 divides 499 & not 7 divides 499 & not 11 divides 499 & not 13 divides 499 & not 17 divides 499 & not 19 divides 499 )
499 = (3 * 166) + 1 ;
hence not 3 divides 499 by NAT_4:9; :: thesis: ( not 5 divides 499 & not 7 divides 499 & not 11 divides 499 & not 13 divides 499 & not 17 divides 499 & not 19 divides 499 )
499 = (5 * 99) + 4 ;
hence not 5 divides 499 by NAT_4:9; :: thesis: ( not 7 divides 499 & not 11 divides 499 & not 13 divides 499 & not 17 divides 499 & not 19 divides 499 )
499 = (7 * 71) + 2 ;
hence not 7 divides 499 by NAT_4:9; :: thesis: ( not 11 divides 499 & not 13 divides 499 & not 17 divides 499 & not 19 divides 499 )
499 = (11 * 45) + 4 ;
hence not 11 divides 499 by NAT_4:9; :: thesis: ( not 13 divides 499 & not 17 divides 499 & not 19 divides 499 )
499 = (13 * 38) + 5 ;
hence not 13 divides 499 by NAT_4:9; :: thesis: ( not 17 divides 499 & not 19 divides 499 )
499 = (17 * 29) + 6 ;
hence not 17 divides 499 by NAT_4:9; :: thesis: not 19 divides 499
499 = (19 * 26) + 5 ;
hence not 19 divides 499 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 499 & n is prime holds
not n divides 499 by XPRIMET1:16;
hence 499 is prime by NAT_4:14; :: thesis: verum