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 Th49:
  MaxADSet(x) = MaxADSet(y) iff Cl {x} = Cl {y}
proof
A1: MaxADSet(x) c= MaxADSet(x) \/ MaxADSet(y) by XBOOLE_1:7;
  thus MaxADSet(x) = MaxADSet(y) implies Cl {x} = Cl {y} by Lm1;
A2: MaxADSet(y) c= MaxADSet(x) \/ MaxADSet(y) by XBOOLE_1:7;
  assume
A3: Cl {x} = Cl {y};
  now
    let a be Point of X;
    assume
A4: a in MaxADSet(x) \/ MaxADSet(y);
    now
      per cases by A4,XBOOLE_0:def 3;
      suppose
        a in MaxADSet(x);
        then
A5:     MaxADSet(a) = MaxADSet(x) by Th21;
        then Cl {a} = Cl {x} by Lm1;
        then
A6:     MaxADSet(y) c= Cl {a} by A3,Th48;
        MaxADSet(x) c= Cl {a} by A5,Th48;
        hence MaxADSet(x) \/ MaxADSet(y) c= Cl {a} by A6,XBOOLE_1:8;
      end;
      suppose
        a in MaxADSet(y);
        then
A7:     MaxADSet(a) = MaxADSet(y) by Th21;
        then Cl {a} = Cl {y} by Lm1;
        then
A8:     MaxADSet(x) c= Cl {a} by A3,Th48;
        MaxADSet(y) c= Cl {a} by A7,Th48;
        hence MaxADSet(x) \/ MaxADSet(y) c= Cl {a} by A8,XBOOLE_1:8;
      end;
    end;
    hence MaxADSet(x) \/ MaxADSet(y) c= Cl {a};
  end;
  then
A9: MaxADSet(x) \/ MaxADSet(y) is anti-discrete;
A10: MaxADSet(y) is maximal_anti-discrete by Th20;
  MaxADSet(x) is maximal_anti-discrete by Th20;
  then MaxADSet(x) = MaxADSet(x) \/ MaxADSet(y) by A9,A1;
  hence thesis by A9,A10,A2;
end;
