reserve x for set,
  R for non empty Poset;

theorem Th1:
  for X,Y being OrderSortedSet of R holds X (/\) Y is OrderSortedSet of R
proof
  let X,Y be OrderSortedSet of R;
  reconsider M = X (/\) Y as ManySortedSet of R;
  M is order-sorted
  proof
    let s1,s2 be Element of R;
    assume s1 <= s2;
    then
A1: X.s1 c= X.s2 & Y.s1 c= Y.s2 by OSALG_1:def 16;
    (X (/\) Y).s1 = X.s1 /\ Y.s1 & (X (/\) Y).s2 = X.s2 /\ Y.s2
      by PBOOLE:def 5;
    hence thesis by A1,XBOOLE_1:27;
  end;
  hence thesis;
end;
