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
  P is closed or Q is closed implies MaxADSet(P /\ Q) = MaxADSet(P) /\
  MaxADSet(Q)
proof
  assume
A1: P is closed or Q is closed;
A2: MaxADSet(P) /\ MaxADSet(Q) c= MaxADSet(P /\ Q)
  proof
    assume not MaxADSet(P) /\ MaxADSet(Q) c= MaxADSet(P /\ Q);
    then consider x being object such that
A3: x in MaxADSet(P) /\ MaxADSet(Q) and
A4: not x in MaxADSet(P /\ Q);
    reconsider x as Point of X by A3;
    now
      per cases by A1;
      suppose
A5:     P is closed;
        then P = MaxADSet(P) by Th60;
        then x in P by A3,XBOOLE_0:def 4;
        then
A6:     MaxADSet(x) c= P by A5,Th23;
A7:     P /\ Q c= MaxADSet(P /\ Q) by Th32;
        x in MaxADSet(Q) by A3,XBOOLE_0:def 4;
        then consider D being set such that
A8:     x in D and
A9:     D in {MaxADSet(b) where b is Point of X : b in Q} by TARSKI:def 4;
        consider b being Point of X such that
A10:    D = MaxADSet(b) and
A11:    b in Q by A9;
        {b} c= MaxADSet(b) by Th12;
        then
A12:    b in MaxADSet(b) by ZFMISC_1:31;
        MaxADSet(x) = MaxADSet(b) by A8,A10,Th21;
        then b in P /\ Q by A6,A11,A12,XBOOLE_0:def 4;
        then MaxADSet(b) meets MaxADSet(P /\ Q) by A12,A7,XBOOLE_0:3;
        then MaxADSet(b) c= MaxADSet(P /\ Q) by Th30;
        hence contradiction by A4,A8,A10;
      end;
      suppose
A13:    Q is closed;
        then Q = MaxADSet(Q) by Th60;
        then x in Q by A3,XBOOLE_0:def 4;
        then
A14:    MaxADSet(x) c= Q by A13,Th23;
A15:    P /\ Q c= MaxADSet(P /\ Q) by Th32;
        x in MaxADSet(P) by A3,XBOOLE_0:def 4;
        then consider D being set such that
A16:    x in D and
A17:    D in {MaxADSet(b) where b is Point of X : b in P} by TARSKI:def 4;
        consider b being Point of X such that
A18:    D = MaxADSet(b) and
A19:    b in P by A17;
        {b} c= MaxADSet(b) by Th12;
        then
A20:    b in MaxADSet(b) by ZFMISC_1:31;
        MaxADSet(x) = MaxADSet(b) by A16,A18,Th21;
        then b in P /\ Q by A14,A19,A20,XBOOLE_0:def 4;
        then MaxADSet(b) meets MaxADSet(P /\ Q) by A20,A15,XBOOLE_0:3;
        then MaxADSet(b) c= MaxADSet(P /\ Q) by Th30;
        hence contradiction by A4,A16,A18;
      end;
    end;
    hence contradiction;
  end;
  MaxADSet(P /\ Q) c= MaxADSet(P) /\ MaxADSet(Q) by Th37;
  hence thesis by A2;
end;
