The argument principle recovers the open mapping theorem

If $f$ is nonconstant and holomorphic on a complex domain and $a\in \mathrm{\Omega }$, then $f-f(a)$ has a zero of positive multiplicity at $a$. Choose a small circle around $a$ on which $f-f(a)$ does not vanish. The preimage-count corollary says that for every $w$ sufficiently close to $f(a)$ the function $f-w$ has the same positive number of zeros inside that circle. So every value near $f(a)$ is attained nearby, and the image is open.

This reproduces the already-published open mapping theorem Open mapping theorem for holomorphic functions. The earlier page keeps its local-normal-form proof because the reading order needs that theorem before the argument principle exists.