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 Th19:
  BDD C misses L~Cage(C,n)
proof
  assume not thesis;
  then consider x being object such that
A1: x in BDD C /\ L~Cage(C,n) by XBOOLE_0:4;
A2: x in L~Cage(C,n) by A1,XBOOLE_0:def 4;
  BDD C = union {B where B is Subset of TOP-REAL 2: B
  is_inside_component_of C} & x in BDD C by A1,JORDAN2C:def 4,XBOOLE_0:def 4;
  then consider Z being set such that
A3: x in Z and
A4: Z in {B where B is Subset of TOP-REAL 2: B is_inside_component_of C}
  by TARSKI:def 4;
  consider B being Subset of TOP-REAL 2 such that
A5: Z = B and
A6: B is_inside_component_of C by A4;
  B misses L~Cage(C,n) by A6,Th18;
  then B /\ L~Cage(C,n) = {};
  hence thesis by A3,A5,A2,XBOOLE_0:def 4;
end;
