theorem Th82:
  D2 c= D1 & the TopStruct of X1 = the TopStruct of X2 implies (D1
  is boundary implies D2 is boundary)
proof
  assume
A1: D2 c= D1;
  then reconsider C1 = D2 as Subset of X1 by XBOOLE_1:1;
  assume
A2: the TopStruct of X1 = the TopStruct of X2;
  assume D1 is boundary;
  then C1 is boundary by A1,Th11;
  then
A3: Int C1 = {};
  Int C1 = Int D2 by A2,Th77;
  hence thesis by A3;
end;
