proposition: Lemma FST_sub_pair T x y x' y' : FSTeq <<= T -> T ⊢ x ≡ x' -> T ⊢ y ≡ y' -> T ⊢ {x; y} ⊆ {x'; y'}.
proof: Proof. intros HT H1 H2. prv_all z. apply II. apply FST_pair_el'; auto. assert1 H. apply FST_adj in H; auto. apply (DE H). - apply DI1. apply (FST_trans') with x; auto. apply (Weak H1). auto. - apply DI2. clear H. assert1 H. apply FST_adj in H; auto. apply (DE H). + apply (FST_trans') with y; auto. apply (Weak H2). auto. + apply Exp. eapply IE; try eapply (FST_eset'); auto. Qed.
imports: Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Deduction.FullNDFacts Undecidability.FOL.Sets.FST Undecidability.FOL.FST Undecidability.FOL.Reductions.PCPb_to_FST Undecidability.PCP(PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb) Undecidability(Shared.ListAutomation)