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 Th53:
  for x, y being Point of X holds MaxADSet(x) misses MaxADSet(y)
iff (ex V being Subset of X st V is open & MaxADSet(x) c= V & V misses MaxADSet
(y)) or ex W being Subset of X st W is open & W misses MaxADSet(x) & MaxADSet(y
  ) c= W
proof
  let x, y be Point of X;
  thus MaxADSet(x) misses MaxADSet(y) implies (ex V being Subset of X st V is
open & MaxADSet(x) c= V & V misses MaxADSet(y)) or ex W being Subset of X st W
  is open & W misses MaxADSet(x) & MaxADSet(y) c= W
  proof
    set V = (Cl {y})`;
    set W = (Cl {x})`;
    assume
A1: MaxADSet(x) /\ MaxADSet(y) = {};
    assume
A2: for V being Subset of X holds V is open & MaxADSet(x) c= V implies
    V meets MaxADSet(y);
    now
      take W;
      thus W is open;
      MaxADSet(x) c= Cl {x} by Th48;
      hence W misses MaxADSet(x) by SUBSET_1:24;
      now
        MaxADSet(y) c= Cl {y} by Th48;
        then V misses MaxADSet(y) by SUBSET_1:24;
        then not MaxADSet(x) c= V by A2;
        then MaxADSet(x) \ V <> {} by XBOOLE_1:37;
        then consider b being object such that
A3:     b in MaxADSet(x) \ V by XBOOLE_0:def 1;
A4:     b in MaxADSet(x) by A3,XBOOLE_0:def 5;
        reconsider b as Point of X by A3;
        not b in V by A3,XBOOLE_0:def 5;
        then b in Cl {y} by SUBSET_1:29;
        then
A5:     {b} c= Cl {y} by ZFMISC_1:31;
        MaxADSet(b) = MaxADSet(x) by A4,Th21;
        then Cl {b} = Cl {x} by Th49;
        then
A6:     Cl {x} c= Cl {y} by A5,TOPS_1:5;
        assume not MaxADSet(y) c= W;
        then MaxADSet(y) \ W <> {} by XBOOLE_1:37;
        then consider a being object such that
A7:     a in MaxADSet(y) \ W by XBOOLE_0:def 1;
A8:     a in MaxADSet(y) by A7,XBOOLE_0:def 5;
        reconsider a as Point of X by A7;
        not a in W by A7,XBOOLE_0:def 5;
        then a in Cl {x} by SUBSET_1:29;
        then
A9:     {a} c= Cl {x} by ZFMISC_1:31;
        MaxADSet(a) = MaxADSet(y) by A8,Th21;
        then Cl {a} = Cl {y} by Th49;
        then Cl {y} c= Cl {x} by A9,TOPS_1:5;
        then Cl {x} = Cl {y} by A6;
        then MaxADSet(x) = MaxADSet(y) by Th49;
        hence contradiction by A1;
      end;
      hence MaxADSet(y) c= W;
    end;
    hence thesis;
  end;
  assume
A10: (ex V being Subset of X st V is open & MaxADSet(x) c= V & V misses
  MaxADSet(y)) or ex W being Subset of X st W is open & W misses MaxADSet(x) &
  MaxADSet(y) c= W;
  assume MaxADSet(x) meets MaxADSet(y);
  then
A11: MaxADSet(x) = MaxADSet(y) by Th22;
  now
    per cases by A10;
    suppose
      ex V being Subset of X st V is open & MaxADSet(x) c= V & V
      misses MaxADSet(y);
      then consider V being Subset of X such that
      V is open and
A12:  MaxADSet(x) c= V and
A13:  V misses MaxADSet(y);
      V /\ MaxADSet(y) = {} by A13;
      hence contradiction by A11,A12,XBOOLE_1:28;
    end;
    suppose
      ex W being Subset of X st W is open & W misses MaxADSet(x) &
      MaxADSet(y) c= W;
      then consider W being Subset of X such that
      W is open and
A14:  W misses MaxADSet(x) and
A15:  MaxADSet(y) c= W;
      W /\ MaxADSet(x) = {} by A14;
      hence contradiction by A11,A15,XBOOLE_1:28;
    end;
  end;
  hence contradiction;
end;
