reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;
reserve x,y for Point of X;
reserve A, B for Subset of X;
reserve P, Q for Subset of X;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace,
  Y0 for non empty SubSpace of X;

theorem
  for X0 being closed SubSpace of X, A being non empty Subset of X holds
  A is Subset of X0 implies MaxADSspace(A) is SubSpace of X0
proof
  let X0 be closed SubSpace of X, A be non empty Subset of X;
  reconsider D = the carrier of X0 as Subset of X by TSEP_1:1;
A1: D is closed by TSEP_1:11;
  assume A is Subset of X0;
  then MaxADSet(A) c= D by A1,Th40;
  then the carrier of MaxADSspace(A) c= the carrier of X0 by Def18;
  hence thesis by Lm2;
end;
