reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th7:
  for A being OrderSortedSet of R, F being ManySortedFunction of
  the carrier of R st F is order-sorted holds F.:.:A is OrderSortedSet of R
proof
  let A be OrderSortedSet of R;
  let F be ManySortedFunction of the carrier of R such that
A1: F is order-sorted;
  reconsider C1 = F.:.:A as ManySortedSet of R;
  C1 is order-sorted
  proof
    let s1,s2 be Element of R;
    assume s1 <= s2;
    then
A2: A.s1 c= A.s2 & F.s1 c= F.s2 by A1,Th1,OSALG_1:def 16;
    C1.s1 = (F.s1).:(A.s1) & C1.s2 = (F.s2).:(A.s2) by PBOOLE:def 20;
    hence thesis by A2,RELAT_1:125;
  end;
  hence thesis;
end;
