  In this paper we provide a proof of unconditional security for a semi-quantum
key distribution protocol introduced in a previous work. This particular
protocol demonstrated the possibility of using $X$ basis states to contribute
to the raw key of the two users (as opposed to using only direct measurement
results) even though a semi-quantum participant cannot directly manipulate such
states. In this work we provide a complete proof of security by deriving a
lower bound of the protocol's key rate in the asymptotic scenario. Using this
bound we are able to find an error threshold value such that for all error
rates less than this threshold, it is guaranteed that $A$ and $B$ may distill a
secure secret key; for error rates larger than this threshold, $A$ and $B$
should abort. We demonstrate that this error threshold compares favorably to
several fully quantum protocols. We also comment on some interesting
observations about the behavior of this protocol under certain noise scenarios.
