reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th4:
  for A being OrderSortedSet of R holds id A is order-sorted
proof
  let A be OrderSortedSet of R;
  set F = id A;
  let s1,s2 be Element of R;
  assume s1 <= s2;
  then
A1: A.s1 c= A.s2 by OSALG_1:def 16;
  let a1 be set such that
A2: a1 in dom (F.s1);
  A.s2 = {} implies A.s2 = {};
  then dom (F.s2) = A.s2 by FUNCT_2:def 1;
  hence a1 in dom (F.s2) by A2,A1;
  (F.s1).a1 = (id (A.s1)).a1 by MSUALG_3:def 1
    .= a1 by A2,FUNCT_1:18
    .= (id (A.s2)).a1 by A2,A1,FUNCT_1:18
    .= (F.s2).a1 by MSUALG_3:def 1;
  hence thesis;
end;
