I am reading the paper "Two Theorems on Extensions of Holomorphic Mappings" by PHILLIP A. GRIFFITHS. In Example 2 of the paper, there is a proposition saying that: Let $N$ be a complex manifold, $S\subset N$ a proper analytic subset, M a projective algebraic variety, and $f: N-S\to M$ a holomorphic map. Then $S$ is a removable singularity for $f$ (in the sense that the closure of the the graph is an analytic subvariety of $N\times M$) if and only if the pull-backs $f^{*}(\varphi)$ of all rational functions $\varphi$ on M extend to meromorphic functions on N.

I do not know why this proposition holds, even if the paper tells me it is obvious.

Any hint is welcomed. Many thanks!