Definition Perp_in_tagged X A B C D := Perp_at X A B C D.

(* Specifications: *)
Lemma Perp_in_Perp_in_tagged : forall X A B C D, Perp_at X A B C D -> Perp_in_tagged X A B C D.
Lemma Perp_in_tagged_Perp_in : forall X A B C D, Perp_in_tagged X A B C D -> Perp_at X A B C D.