reserve i, j, k, n for Nat,
  P for Subset of TOP-REAL 2,
  C for connected compact non vertical non horizontal Subset of TOP-REAL 2;

theorem Th16:
  BDD C misses LeftComp Cage(C,n)
proof
  set f = Cage(C,n);
  assume BDD C /\ LeftComp f <> {};
  then consider x being Point of TOP-REAL 2 such that
A1: x in BDD C /\ LeftComp f by SUBSET_1:4;
  BDD C misses UBD C by JORDAN2C:24;
  then
A2: BDD C /\ UBD C = {};
  x in BDD C by A1,XBOOLE_0:def 4;
  then not x in UBD C by A2,XBOOLE_0:def 4;
  then
A3: not x in union UBD-Family C by Th14;
A4: x in LeftComp f by A1,XBOOLE_0:def 4;
  UBD L~f in the set of all  UBD L~Cage(C,k) where k is Nat ;
  then not x in UBD L~f by A3,TARSKI:def 4;
  hence contradiction by A4,GOBRD14:36;
end;
