The Dolecki-Greco-Lechicki Theorem

I have met Szymon Dolecki at the Summer Topology Conference 2016, and he is a charming person.  In 1995, with Gabriele Greco and Alojzy Lechicki, he proved a very nice theorem — every Čech-complete space is consonant — that deserves to be well-known.  As a coincidence, this very theorem was mentioned by Jimmie Lawson at the same conference.  It is also the subject of Exercise 8.3.4 in the book, but I am afraid that, as stated, it is way too hard. My purpose here is to give a complete solution to the exercise.  The proof is elementary, that is, it does not require any deep knowledge of topology.  Read the full post.