Prove that the K-K-M theorem is equivalent to Brouwer's theorem, that is, K-K-M theorem Brouwer's theorem

Question:

Prove that the K-K-M theorem is equivalent to Brouwer's theorem, that is,
K-K-M theorem ⇔ Brouwer's theorem
The classic application of the Brouwer theorem in economics is to prove the existence of competitive equilibrium. We extract the mathematical essence in the following corollary, and then show how it applies to competitive equilibrium in example 2.95.