Corollary 1

Suppose that F is separated. Let $v\in \mathscr{L}^f(E; F)$. If $u$ is a strict morphism with closed image from E into F, then the same is true of $u+v$.

Since F is separated, the kernel of $v$ is a closed vector subspace of E; it is of finite codimension in E and $u+v$ has the same restriction as $u$ to Ker($v$). The corollary follows thus from the proposition.