now :: thesis: ( not 2 divides 449 & not 3 divides 449 & not 5 divides 449 & not 7 divides 449 & not 11 divides 449 & not 13 divides 449 & not 17 divides 449 & not 19 divides 449 )

449 = (2 * 224) + 1 ;
hence not 2 divides 449 by NAT_4:9; :: thesis: ( not 3 divides 449 & not 5 divides 449 & not 7 divides 449 & not 11 divides 449 & not 13 divides 449 & not 17 divides 449 & not 19 divides 449 )
449 = (3 * 149) + 2 ;
hence not 3 divides 449 by NAT_4:9; :: thesis: ( not 5 divides 449 & not 7 divides 449 & not 11 divides 449 & not 13 divides 449 & not 17 divides 449 & not 19 divides 449 )
449 = (5 * 89) + 4 ;
hence not 5 divides 449 by NAT_4:9; :: thesis: ( not 7 divides 449 & not 11 divides 449 & not 13 divides 449 & not 17 divides 449 & not 19 divides 449 )
449 = (7 * 64) + 1 ;
hence not 7 divides 449 by NAT_4:9; :: thesis: ( not 11 divides 449 & not 13 divides 449 & not 17 divides 449 & not 19 divides 449 )
449 = (11 * 40) + 9 ;
hence not 11 divides 449 by NAT_4:9; :: thesis: ( not 13 divides 449 & not 17 divides 449 & not 19 divides 449 )
449 = (13 * 34) + 7 ;
hence not 13 divides 449 by NAT_4:9; :: thesis: ( not 17 divides 449 & not 19 divides 449 )
449 = (17 * 26) + 7 ;
hence not 17 divides 449 by NAT_4:9; :: thesis: not 19 divides 449
449 = (19 * 23) + 12 ;
hence not 19 divides 449 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 449 & n is prime holds
not n divides 449 by XPRIMET1:16;
hence 449 is prime by NAT_4:14; :: thesis: verum