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;

theorem Th47:
  MaxADSet(x) c= meet {F where F is Subset of X : F is closed & x in F}
proof
  set G = {F where F is Subset of X : F is closed & x in F};
  [#]X in G; then
  A1: G <> {};
  G c= bool the carrier of X
  proof
    let C be object;
    assume C in G;
    then ex P being Subset of X st C = P & P is closed & x in P;
    hence thesis;
  end;
  then reconsider G as Subset-Family of X;
  now
    let C be set;
    assume C in G;
    then ex F being Subset of X st F = C & F is closed & x in F;
    hence MaxADSet(x) c= C by Th23;
  end;
  hence thesis by A1,SETFAM_1:5;
end;
