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 Th11:
  C c= RightComp Cage(C,n)
proof
  set f = Cage(C,n);
  let c be object;
  assume
A1: c in C;
  C misses L~f by Th5;
  then C /\ L~f = {};
  then
A2: not c in L~f by A1,XBOOLE_0:def 4;
  C misses LeftComp f by Th10;
  then C /\ LeftComp f = {};
  then not c in LeftComp f by A1,XBOOLE_0:def 4;
  hence thesis by A1,A2,GOBRD14:18;
end;
