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

theorem Th52:
  X (\) Y = EmptyMS I iff X c= Y
proof
  hereby
    assume
A1: X (\) Y = EmptyMS I;
    now
      let i be object;
      assume i in I;
      then X.i \ Y.i = (X (\) Y).i by Def6
        .= {} by A1;
      hence X.i c= Y.i by XBOOLE_1:37;
    end;
    hence X c= Y;
  end;
  assume
A2: X c= Y;
  now
    let i be object;
    assume
A3: i in I;
    then
A4: X.i c= Y.i by A2;
    thus (X (\) Y).i = X.i \ Y.i by A3,Def6
      .= {} by A4,XBOOLE_1:37;
  end;
  hence thesis by Th6;
end;
