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 Th31:
  for A, B being Subset of Y holds A c= B implies MaxADSet(A) c= MaxADSet(B)
proof
  let A, B be Subset of Y;
  set E = {MaxADSet(a) where a is Point of Y : a in A};
  set F = {MaxADSet(b) where b is Point of Y : b in B};
  assume
A1: A c= B;
  E c= F
  proof
    let C be object;
    assume C in E;
    then ex a being Point of Y st C = MaxADSet(a) & a in A;
    hence thesis by A1;
  end;
  hence thesis by ZFMISC_1:77;
end;
