theorem
  C is connected & i <= j implies RightComp(Cage(C,j)) c= RightComp(Cage
  (C, i ) )
proof
  assume C is connected & i <= j;
  then
A1: Cl LeftComp(Cage(C,i)) c= Cl LeftComp(Cage(C,j)) by Th47,PRE_TOPC:19;
  (Cl LeftComp(Cage(C,i)))` = RightComp(Cage(C,i)) & (Cl LeftComp(Cage(C,j
  )))` = RightComp(Cage(C,j)) by Th42;
  hence thesis by A1,SUBSET_1:12;
end;
