I have a problem in the proof of point c of theorem 15.1.9 (Rudin, Function theory of the unit ball of $C^n$ enter image description here).
In particular in one passage it is necessary to prove that, if $F$ is a proper holomorphic function between connected open sets of $C^n, A, A'$, and $V$ is the locus of the zeros of a $g$, holomorphic function in $A$, then $F(V)$ is the locus of the zeros of a function h,holomorphic in $A'$.
To do this he defines a suitable holomorphic map h in the regular values of $F$ with the desired property and proves that it can be extended holomorphically to all $A'$.
here unfortunately I stop understanding, to prove that it is extensible in the desired way he proves that, taking an arbitrary compact of $A'$, $h$ restricted to the intersection between $K$ and the regular values of $F$ is bounded, he concludes that therefore h is extensible because the set of critical values is $H^\infty$ removable. Why, given this property, should h be extensible?