Definition AnnIso_respects_atoms_eq   := third  ann_respects_atoms_eq_mutual.Definition AnnDefEq_respects_atoms_eq := fourth ann_respects_atoms_eq_mutual.

Lemma ann_remove_available_mutual: (forall G1 a A, AnnTyping G1 a A -> True) /\ (forall G1 phi, AnnPropWff G1 phi -> True) /\ (forall G1 D g p1 p2, AnnIso G1 D g p1 p2 -> AnnIso G1 (AtomSetImpl.inter D (dom G1)) g p1 p2) /\ (forall G1 D g A B, AnnDefEq G1 D g A B -> AnnDefEq G1 (AtomSetImpl.inter D (dom G1)) g A B) /\ (forall G1 , AnnCtx G1 -> True).