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 non empty SubSpace of X, x being Point of X holds
  x is Point of X0 implies MaxADSspace(x) is SubSpace of X0
proof
  let X0 be closed non empty SubSpace of X, x be Point of X;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
A1: A is closed by TSEP_1:11;
  assume x is Point of X0;
  then MaxADSet(x) c= A by A1,Th23;
  then the carrier of MaxADSspace(x) c= the carrier of X0 by Def17;
  hence thesis by Lm2;
end;
