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
  MaxADSet(A) = MaxADSet(B) implies Cl A = Cl B
proof
A1: MaxADSet(A) c= Cl A by Th59;
A2: MaxADSet(B) c= Cl B by Th59;
  assume
A3: MaxADSet(A) = MaxADSet(B);
  then A c= MaxADSet(B) by Th32;
  then A c= Cl B by A2;
  then
A4: Cl A c= Cl B by TOPS_1:5;
  B c= MaxADSet(A) by A3,Th32;
  then B c= Cl A by A1;
  then Cl B c= Cl A by TOPS_1:5;
  hence thesis by A4;
end;
