How to Prove a Lemma in Lambda Calculus about Contexts

I don't remember ever seeing a definition of a context to include a variable. A plain variable does not have a hole, after all. But, moving on...

$F$ does not need to be well-scoped. So, for the base case where $C=y$, you can define $F$ to be $\lambda f.y$.

For the next base case, where $C[\,]=[\,]$, you can define $F$ to be the function $\lambda f. f\,\vec{x}$, where $\vec{x}$ is the $\vec{x}$ given by the statement. Then $C[M] = M$ and $F\,(\lambda\vec{x}.M) = (\lambda f. f\,\vec{x})\,(\lambda\vec{x}.M)$.

Note that in both cases, the two terms are not syntactically equal, but they are $\beta$-equivalent.