reserve i,j,e,u for object;
reserve I for set; 
reserve x,X,Y,Z,V for ManySortedSet of I;

theorem Th8:
  x in X (/\) Y iff x in X & x in Y
proof
  hereby
    assume
A1: x in X (/\) Y;
    thus x in X
    proof
      let i be object;
      assume
A2:   i in I;
      then x.i in (X (/\) Y).i by A1;
      then x.i in X.i /\ Y.i by A2,Def5;
      hence thesis by XBOOLE_0:def 4;
    end;
    thus x in Y
    proof
      let i be object;
      assume
A3:   i in I;
      then x.i in (X (/\) Y).i by A1;
      then x.i in X.i /\ Y.i by A3,Def5;
      hence thesis by XBOOLE_0:def 4;
    end;
  end;
  assume
A4: x in X & x in Y;
  let i be object;
  assume
A5: i in I;
  then x.i in X.i & x.i in Y.i by A4;
  then x.i in X.i /\ Y.i by XBOOLE_0:def 4;
  hence thesis by A5,Def5;
end;
