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 Th18:
  P is_inside_component_of C implies P misses L~Cage(C,n)
proof
  set f = Cage(C,n);
  assume P is_inside_component_of C;
  then
A1: P c= BDD C by JORDAN2C:22;
  assume P /\ L~f <> {};
  then consider x being Point of TOP-REAL 2 such that
A2: x in P /\ L~f by SUBSET_1:4;
  x in P by A2,XBOOLE_0:def 4;
  then
A3: x in BDD C by A1;
A4: BDD C c= RightComp f by Th17;
  x in L~f by A2,XBOOLE_0:def 4;
  hence contradiction by A3,A4,GOBRD14:18;
end;
