Rockafellar theorem 7.4 - Agreement of cl f with f within the relative interior of dom f

Question

I am trying to make sense of Theorem 7.4 in Rockafellar's "Convex Analysis"

After applying lemma 7.3, Rockafellar uses Corollary 6.5.1 to establish equality between three intersections. I understand that the first equality comes directly from Corollary 6.5.1, but how is the second equality obtained? Why can the closure operation be completely discounted?

I have left relevant proven statements below.

EDIT: Does this equality arise trivially from the strict inequality in the definition of the epigraph?

Answer 1

Indeed it follows from the definition of the epigraph (which does not have a strict inequality). For a given $x \in \text{ri dom} f$, you get: $$M \cap \text{epi} f = \{(x,\mu) : f(x) \leq \mu \},$$ which is closed.