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 Th46:
  MaxADSet(x) c= meet {G where G is Subset of X : G is open & x in G}
proof
  set F = {G where G is Subset of X : G is open & x in G};
  [#]X in F; then
  A1: F <> {};
  F c= bool the carrier of X
  proof
    let C be object;
    assume C in F;
    then ex P being Subset of X st C = P & P is open & x in P;
    hence thesis;
  end;
  then reconsider F as Subset-Family of X;
  now
    let C be set;
    assume C in F;
    then ex G being Subset of X st G = C & G is open & x in G;
    hence MaxADSet(x) c= C by Th24;
  end;
  hence thesis by A1,SETFAM_1:5;
end;
