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 Th10:
  C misses LeftComp Cage(C,n)
proof
  set f = Cage(C,n);
  assume
A1: C meets LeftComp f;
  RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2;
  then
A2: ex R being Subset of (TOP-REAL 2)|(L~f)` st R = RightComp f & R
  is a_component by CONNSP_1:def 6;
  C misses L~f by Th5;
  then
A3: C /\ L~f = {};
  C c= the carrier of (TOP-REAL 2)|(L~f)`
  proof
    let c be object;
    assume
A4: c in C;
    then not c in L~f by A3,XBOOLE_0:def 4;
    then c in (L~f)` by A4,SUBSET_1:29;
    hence thesis by PRE_TOPC:8;
  end;
  then reconsider C1 = C as Subset of (TOP-REAL 2)|(L~f)`;
  LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1;
  then
A5: ex L being Subset of (TOP-REAL 2)|(L~f)` st L = LeftComp f & L
  is a_component by CONNSP_1:def 6;
  C meets RightComp f & C1 is connected by Th9,CONNSP_1:23;
  hence contradiction by A1,A5,A2,JORDAN2C:92,SPRECT_4:6;
end;
