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 Th5:
  C misses L~Cage(C,n)
proof
  set f = Cage(C,n), G = Gauge(C,n);
  assume not thesis;
  then consider c being object such that
A1: c in C and
A2: c in L~f by XBOOLE_0:3;
  L~f = union { LSeg(f,i) where i is Nat: 1 <= i & i+1 <= len f
  } by TOPREAL1:def 4;
  then consider Z being set such that
A3: c in Z and
A4: Z in { LSeg(f,i) where i is Nat: 1 <= i & i+1 <= len f }
  by A2,TARSKI:def 4;
  consider i being Nat such that
A5: Z = LSeg(f,i) and
A6: 1 <= i & i+1 <= len f by A4;
  f is_sequence_on G by JORDAN9:def 1;
  then LSeg(f,i) = left_cell(f,i,G) /\ right_cell(f,i,G) by A6,GOBRD13:29;
  then
A7: c in left_cell(f,i,G) by A3,A5,XBOOLE_0:def 4;
  left_cell(f,i,G) misses C by A6,JORDAN9:31;
  hence contradiction by A1,A7,XBOOLE_0:3;
end;
