theorem
  for X0 being open 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 open 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 open by TSEP_1:16;
  assume x is Point of X0;
  then MaxADSet(x) c= A by A1,Th24;
  then the carrier of MaxADSspace(x) c= the carrier of X0 by Def17;
  hence thesis by Lm2;
end;
