general topology - Bourbaki on the fact that continuous function on a compact is uniformly continuous

I am now looking theorem 2 in paragraph 4.1 of:

Bourbaki. "Elements of Mathematics General Topology. Part 1".

THEOREM 2. Every continuous mapping $f$ of a compact space $X$ into a uniform space $X'$ is uniformly continuous.

Now I realize that I don't understand the proof.

Could you present me a more detailed proof?