let X be non empty TopSpace; :: thesis: for x, y being Point of X holds
( MaxADSet x = MaxADSet y iff Cl {x} = Cl {y} )
let x, y be Point of X; :: thesis: ( MaxADSet x = MaxADSet y iff Cl {x} = Cl {y} )
A1: MaxADSet x c= (MaxADSet x) \/ (MaxADSet y) by XBOOLE_1:7;
thus ( MaxADSet x = MaxADSet y implies Cl {x} = Cl {y} ) by Lm1; :: thesis: ( Cl {x} = Cl {y} implies MaxADSet x = MaxADSet y )
A2: MaxADSet y c= (MaxADSet x) \/ (MaxADSet y) by XBOOLE_1:7;
assume A3: Cl {x} = Cl {y} ; :: thesis: MaxADSet x = MaxADSet y

now

let a be Point of X; :: thesis: ( a in (MaxADSet x) \/ (MaxADSet y) implies (MaxADSet x) \/ (MaxADSet y) c= Cl {a} )
assume A4: a in (MaxADSet x) \/ (MaxADSet y) ; :: thesis: (MaxADSet x) \/ (MaxADSet y) c= Cl {a}

now

per cases ( a in MaxADSet x or a in MaxADSet y ) by A4, XBOOLE_0:def 3;

suppose a in MaxADSet x ; :: thesis: (MaxADSet x) \/ (MaxADSet y) c= Cl {a}

then A5: MaxADSet a = MaxADSet x by Th23;
then Cl {a} = Cl {x} by Lm1;
then A6: MaxADSet y c= Cl {a} by A3, Th50;
MaxADSet x c= Cl {a} by A5, Th50;
hence (MaxADSet x) \/ (MaxADSet y) c= Cl {a} by A6, XBOOLE_1:8; :: thesis: verum

end;

suppose a in MaxADSet y ; :: thesis: (MaxADSet x) \/ (MaxADSet y) c= Cl {a}

then A7: MaxADSet a = MaxADSet y by Th23;
then Cl {a} = Cl {y} by Lm1;
then A8: MaxADSet x c= Cl {a} by A3, Th50;
MaxADSet y c= Cl {a} by A7, Th50;
hence (MaxADSet x) \/ (MaxADSet y) c= Cl {a} by A8, XBOOLE_1:8; :: thesis: verum

end;

end;

end;

hence (MaxADSet x) \/ (MaxADSet y) c= Cl {a} ; :: thesis: verum

end;

then A9: (MaxADSet x) \/ (MaxADSet y) is anti-discrete by Def12;
A10: MaxADSet y is maximal_anti-discrete by Th22;
MaxADSet x is maximal_anti-discrete by Th22;
then MaxADSet x = (MaxADSet x) \/ (MaxADSet y) by A9, A1, Def7;
hence MaxADSet x = MaxADSet y by A9, A10, A2, Def7; :: thesis: verum