Let 
denote the encoding of TM M, and let 
denote the encoding of string u. It is known that the language
is not Turing-decidable. From this, however, is doesn't follow that if we index the machine-input strings to a paticular machine M the resulting set
is not Turing-decidable. However, we do know that
is not Turing-decidable, where 
is a TM accepting
. Our second theorem above follows from this fact. (Make sure
you can verify it.)