reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;
reserve X0 for SubSpace of X;
reserve X0 for non empty SubSpace of X;
reserve X1,X2 for TopStruct;
reserve X1,X2 for TopSpace;
reserve D1 for Subset of X1,
  D2 for Subset of X2;

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;
