now :: thesis: ( not 2 divides 419 & not 3 divides 419 & not 5 divides 419 & not 7 divides 419 & not 11 divides 419 & not 13 divides 419 & not 17 divides 419 & not 19 divides 419 )

419 = (2 * 209) + 1 ;
hence not 2 divides 419 by NAT_4:9; :: thesis: ( not 3 divides 419 & not 5 divides 419 & not 7 divides 419 & not 11 divides 419 & not 13 divides 419 & not 17 divides 419 & not 19 divides 419 )
419 = (3 * 139) + 2 ;
hence not 3 divides 419 by NAT_4:9; :: thesis: ( not 5 divides 419 & not 7 divides 419 & not 11 divides 419 & not 13 divides 419 & not 17 divides 419 & not 19 divides 419 )
419 = (5 * 83) + 4 ;
hence not 5 divides 419 by NAT_4:9; :: thesis: ( not 7 divides 419 & not 11 divides 419 & not 13 divides 419 & not 17 divides 419 & not 19 divides 419 )
419 = (7 * 59) + 6 ;
hence not 7 divides 419 by NAT_4:9; :: thesis: ( not 11 divides 419 & not 13 divides 419 & not 17 divides 419 & not 19 divides 419 )
419 = (11 * 38) + 1 ;
hence not 11 divides 419 by NAT_4:9; :: thesis: ( not 13 divides 419 & not 17 divides 419 & not 19 divides 419 )
419 = (13 * 32) + 3 ;
hence not 13 divides 419 by NAT_4:9; :: thesis: ( not 17 divides 419 & not 19 divides 419 )
419 = (17 * 24) + 11 ;
hence not 17 divides 419 by NAT_4:9; :: thesis: not 19 divides 419
419 = (19 * 22) + 1 ;
hence not 19 divides 419 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 419 & n is prime holds
not n divides 419 by XPRIMET1:16;
hence 419 is prime by NAT_4:14; :: thesis: verum