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 Th23:
  for F being Subset of Y, x being Point of Y st F is closed & x
  in F holds MaxADSet(x) c= F
proof
  let F be Subset of Y, x be Point of Y;
  assume that
A1: F is closed and
A2: x in F;
  {x} c= MaxADSet(x) by Th12;
  then
A3: x in MaxADSet(x) by ZFMISC_1:31;
  MaxADSet(x) is maximal_anti-discrete by Th20;
  then MaxADSet(x) is anti-discrete;
  hence thesis by A1,A2,A3;
end;
