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;
reserve A, B for Subset of X;
reserve P, Q for Subset of X;

theorem Th56:
  P is open implies MaxADSet(P) = P
proof
  set F = {G where G is Subset of X : G is open & P c= G};
A1: P c= MaxADSet(P) by Th32;
  assume P is open;
  then P in F;
  then
A2: meet F c= P by SETFAM_1:3;
  MaxADSet(P) c= meet F by Th55;
  then MaxADSet(P) c= P by A2;
  hence thesis by A1;
end;
