reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;

theorem
  for A, B being Subset of Y holds MaxADSet(A \/ B) = MaxADSet(A) \/
  MaxADSet(B)
proof
  let A, B be Subset of Y;
A1: MaxADSet(A \/ B) c= MaxADSet(A) \/ MaxADSet(B)
  proof
    let x be object;
    assume x in MaxADSet(A \/ B);
    then consider C being set such that
A2: x in C and
A3: C in {MaxADSet(a) where a is Point of Y : a in A \/ B} by TARSKI:def 4;
    consider a being Point of Y such that
A4: C = MaxADSet(a) and
A5: a in A \/ B by A3;
    now
      per cases by A5,XBOOLE_0:def 3;
      suppose
A6:     a in A;
        now
          take C;
          thus x in C by A2;
          thus C in {MaxADSet(c) where c is Point of Y : c in A} by A4,A6;
        end;
        then
A7:     x in MaxADSet(A) by TARSKI:def 4;
        MaxADSet(A) c= MaxADSet(A) \/ MaxADSet(B) by XBOOLE_1:7;
        hence thesis by A7;
      end;
      suppose
A8:     a in B;
        now
          take C;
          thus x in C by A2;
          thus C in {MaxADSet(c) where c is Point of Y : c in B} by A4,A8;
        end;
        then
A9:     x in MaxADSet(B) by TARSKI:def 4;
        MaxADSet(B) c= MaxADSet(A) \/ MaxADSet(B) by XBOOLE_1:7;
        hence thesis by A9;
      end;
    end;
    hence thesis;
  end;
A10: MaxADSet(B) c= MaxADSet(A \/ B) by Th31,XBOOLE_1:7;
  MaxADSet(A) c= MaxADSet(A \/ B) by Th31,XBOOLE_1:7;
  then MaxADSet(A) \/ MaxADSet(B) c= MaxADSet(A \/ B) by A10,XBOOLE_1:8;
  hence thesis by A1;
end;
