reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;

theorem Th38:
  for G being Subset of Y, A being Subset of Y st G is open & A c=
  G holds MaxADSet(A) c= G
proof
  let G be Subset of Y, A be Subset of Y;
  assume
A1: G is open;
  assume
A2: A c= G;
  MaxADSet(A) c= G
  proof
    let x be object;
    assume
A3: x in MaxADSet(A);
    then reconsider a = x as Point of Y;
    consider D being set such that
A4: a in D and
A5: D in {MaxADSet(b) where b is Point of Y : b in A} by A3,TARSKI:def 4;
    consider b being Point of Y such that
A6: D = MaxADSet(b) and
A7: b in A by A5;
A8: MaxADSet(a) = MaxADSet(b) by A4,A6,Th21;
A9: {a} c= MaxADSet(a) by Th12;
    MaxADSet(b) c= G by A1,A2,A7,Th24;
    then {a} c= G by A8,A9;
    hence thesis by ZFMISC_1:31;
  end;
  hence thesis;
end;
