Degree of a line bundle using the zero locus of a holomorphic section

Question

I am reading Kobayashi's book DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES and in the proof of its Lemma 5.7.5, he uses the fact that when $L \rightarrow M$ is a complex holomorphic line bundle over a compact Kähler manifold $M$ of dimension $n$ with Kähler form $\Phi$ and $s : M \rightarrow L$ is a non-zero holomorphic section, then, $$ \deg(L) = \int_M c_1(L) \land \Phi^{n - 1} = \int_V \Phi^{n - 1}, $$ where $V = \{s = 0\}$ is the zero locus divisor (I guess zeroes are counted with multiplicities). He uses it to prove that any line bundle that admits a holomorphic section has a non-negative degree and I am pretty sure the formula $$ \int_M c_1(L) \land \Phi^{n - 1} = \int_{\{s = 0\}} \Phi^{n - 1} $$ remains true when $s$ is a meromorphic function (in this case, the integral can have a negative part), but I don't understand how we get this formula. Can someone give me a proof or a reference for a proof please ?