I was reading the following paper by Jim Kadin, $P^{NP[O(\text{log }  n)]}$ and sparse Turing complete sets for NPThe main result is that if there is a sparse set $S \in NP$ such that $coNP \subseteq NP^S $, then $PH \subseteq P^{SAT[O(\text{log }n)]} $.In the proof of this result he author claims that it is sufficient to prove $NP^ {SAT} \subseteq P^{SAT[O(\text{log }n)]}$ and this will imply that $PH \subseteq P^{SAT[O(\text{log }n)]} $. I could not figure out why this is true. Perhaps this is trivial and there is some minor point (some inclusion of complexity classes) that I am missing. Please do point out why the above claim is true. 
Why would $NP^ {SAT} \subseteq P^{SAT[O(\text{log }n)]}$ imply that $PH \subseteq P^{SAT[O(\text{log }n)]} $
complexity theory;turing machines;oracle machines