Section EXTRA1.

Lemma extend_alpha_proves {L : language} (Gamma : ensemble (frm L)) (Gamma' : ensemble (frm L)) (C : frm L)
(SUBSET : forall p : frm L, forall IN : p \in Gamma, exists p' : frm L, p ≡ p' /\ p' \in Gamma')
(PROVE : Gamma ⊢ C)
: Gamma' ⊢ C.