Def: Suppose X is topological space and B is a base for it. We say, that B is normal base, if following properties hold:

a. For any x∈X and A∈B, with x∈A, there exist A′∈B, such that x∉A′ and A∪A′=X.

b. If U and V are open sets from B, such that U∪V=X, than there exist U′ and V′, disjoint sets from B, such that X∖U⊆V′ and X∖V⊆U′.

It is well known that the space is completely regular iff it has normal base. This characterizations first published in Orrin Frink's paper "compactifications and semi-normal spaces".

It's easy part to prove that every completely regular space has normal base, because the set of co-zero sets is such a base, which is well-known to be a base for a completely regular space.

Inverse part, that every space which has normal base is completely regular, maybe is also easy, because there is a hint that this is actually the rewriting of proof of Urysohn lemma. But, however I still have a problem with this proof and can't construct the scheme like from Urysohn's lemma. Please, help me if you can.