I am studying some fixed point theory, and bumped into this Lemma from which Kakutani Fixed Point Theorem is claimed to follow trivially:
Lemma
Let $K\subset\mathbb{R}^m$ be compact and convex and let $\mu:K\twoheadrightarrow K$. Suppose there are:
-A closed correspondence $\gamma:K\twoheadrightarrow F$ with nonempty convex values where $F\subset\mathbb{R}^k$ is compact and convex.
-A continuous map $f:K\times F\rightarrow K$ such that $\forall x\in K\ \mu(x)=\{f(x,y):\ y\in\gamma(x)\}$
Then $\mu$ has a fixed point.
Kakutani's Theorem
Let $K\subset\mathbb{R}^m$ be compact and convex and $\psi:K\twoheadrightarrow K$ be closed or upper hemicontinuous with nonempty convex compact values. Then $\psi$ has a fixed point.
My question is: in order to prove Kakutani's is sufficient to show:
1)Upper hemicontinuous with compact values $\Rightarrow$ closed (if not already closed by hypothesis)
2)Enforce the Lemma by choosing $F=K$, $\mu=\gamma=\psi$, $f(x,y)=x$ the continuous projection on the first factor ?
If not, how to see the implication?