reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;

theorem
  subset-closed_closure_of (X\/Y) =
    subset-closed_closure_of X \/ subset-closed_closure_of Y
 proof
  set fxy=subset-closed_closure_of(X\/Y);
  set fx=subset-closed_closure_of X;
  set fy=subset-closed_closure_of Y;
  hereby let x be object;
    reconsider xx=x as set by TARSKI:1;
   assume x in fxy;
   then consider y such that
    A1: xx c=y and
    A2: y in X\/Y by Th2;
   y in X or y in Y by A2,XBOOLE_0:def 3;
   then x in fx or x in fy by A1,Th2;
   hence x in fx\/fy by XBOOLE_0:def 3;
  end;
  let x be object;
    reconsider xx=x as set by TARSKI:1;
  assume A3: x in fx\/fy;
  per cases by A3,XBOOLE_0:def 3;
  suppose x in fx;
   then consider y such that
    A4: xx c=y and
    A5: y in X by Th2;
   y in X\/Y by A5,XBOOLE_0:def 3;
   hence thesis by A4,Th2;
  end;
  suppose x in fy;
   then consider y such that
    A6: xx c=y and
    A7: y in Y by Th2;
   y in X\/Y by A7,XBOOLE_0:def 3;
   hence thesis by A6,Th2;
  end;
 end;
