Question about the Uniqueness of representation theorem for open sets on $\mathbb{R}$

This is from the book $<<$Fundamentals of Real Analysis$>>$ by Sterling K Berberian, Section 5.7 F.Riesz's "Rising Sun Lemma".

The author first proves a lemma (The author calls it Riesz's Lemma) :

and in Exercise 1:

Question:Although the author has given me hints on how to solve this question, I have to admit that I dont know how to even start. I saw these two post Uniqueness of Representation for Open Sets on the Real Line and Representation Theorem for Open Sets on The Real Line (Proof-explanation). These two post are related to the question of mine but both of them does not solve my question using the authors hint (at least for now, but maybe i had mistaken). How do I solve this question using the given hint?