Definition Iso_respects_atoms_eq   := third  respects_atoms_eq_mutual.Definition DefEq_respects_atoms_eq := fourth respects_atoms_eq_mutual.

Lemma remove_available_mutual: (forall G1 a A, Typing G1 a A -> True) /\ (forall G1 phi, PropWff G1 phi -> True) /\ (forall G1 D p1 p2, Iso G1 D p1 p2 -> Iso G1 (AtomSetImpl.inter D (dom G1)) p1 p2) /\ (forall G1 D A B T, DefEq G1 D A B T -> DefEq G1 (AtomSetImpl.inter D (dom G1)) A B T) /\ (forall G1 , Ctx G1 -> True).