Prove that the no-retraction theorem is equivalent to Brouwer's theorem. The following proposition, due to Knaster, Kuratowki,

Question:

Prove that the no-retraction theorem is equivalent to Brouwer's theorem.
The following proposition, due to Knaster, Kuratowki, and Mazurkiewicz (K-K-M), is equivalent to the Brouwer theorem. It is often used as a step on the way to the Brouwer theorem. It is more useful than the Brouwer theorem in some applications.