Splitting lemma, a detail

Question

Let R be a commutative unitary ring and let $0\overset{}{\to}N \overset{f}{\to} M \overset{g}{\to} P \to 0$ a short exact sequence of R-modules. Now suppose that the sequence splits. If N and P are submodules of M, then N and P are in direct INNER sum, i.e. $M=N+P$ and $N\cap P=\{0\}$: this sentence in bold is true (I think), how I can prove it in a rigorous Way?