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 Th17:
  BDD C c= RightComp Cage(C,n)
proof
  set f = Cage(C,n);
  let x be object;
  LeftComp f is_outside_component_of L~f by GOBRD14:34;
  then
A1: UBD L~f = union {E where E is Subset of TOP-REAL 2: E
is_outside_component_of L~f} & LeftComp f in {E where E is Subset of TOP-REAL 2
  : E is_outside_component_of L~f} by JORDAN2C:def 5;
  assume
A2: x in BDD C;
A3: now
    assume x in Cl LeftComp f;
    then BDD C meets LeftComp f by A2,PRE_TOPC:def 7;
    hence contradiction by Th16;
  end;
  BDD C misses UBD C by JORDAN2C:24;
  then BDD C /\ UBD C = {};
  then not x in UBD C by A2,XBOOLE_0:def 4;
  then
A4: not x in union UBD-Family C by Th14;
  UBD L~f in the set of all  UBD L~Cage(C,k) where k is Nat ;
  then not x in UBD L~Cage(C,n) by A4,TARSKI:def 4;
  then
A5: not x in LeftComp f by A1,TARSKI:def 4;
  L~f = (Cl LeftComp f) \ LeftComp f by GOBRD14:20;
  then not x in L~f by A3,XBOOLE_0:def 5;
  hence thesis by A2,A5,GOBRD14:18;
end;
