theorem
  A c= B & x is_an_accumulation_point_of A implies x
  is_an_accumulation_point_of B
proof
  assume A c= B;
  then
A1: Der A c= Der B by Th30;
  assume x is_an_accumulation_point_of A;
  then x in Der A by Th16;
  hence thesis by A1,Th16;
end;
