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;

theorem
  for X0 being open non empty SubSpace of X for A being Subset of X, B
  being Subset of X0 st A = B holds A is boundary iff B is boundary
proof
  let X0 be open non empty SubSpace of X;
  let A be Subset of X, B be Subset of X0;
  reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
A1: C is open by TSEP_1:def 1;
  assume A = B;
  hence thesis by A1,Th65,Th66;
end;
