for C being compact Subset of (TOP-REAL 2) holds
( proj2 .: (C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is closed & proj2 .: (C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is bounded_below & proj2 .: (C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2))) is bounded_above )