It is a well-known result in functional analysis that the sum $M+N$ of two subspaces of a Banach space with $M\cap N=0$ is closed if and only if the inclination $$\widehat{(M,N)} := \inf_{x\in M, \|x\|=1} d(x,N)$$ is positive, i.e. $$ M+N \text{closed} \Leftrightarrow \widehat{(M,N)}>0.$$ Typically this is quoted from

```
Gurariĭ, V. I.: Openings and inclinations of subspaces of a Banach space. Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 1 1965 194–204.
```

where this is stated as a Theorem but without proof.

I am looking for a reference containing a proof of this result.