reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

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;
