reserve x for set,
  R for non empty Poset;
reserve S1 for OrderSortedSign,
  OU0 for OSAlgebra of S1;
reserve s,s1,s2,s3,s4 for SortSymbol of S1;

theorem Th17:
  for A being OSSubset of OU0 holds the Sorts of OU0 in OSSubSort( A)
proof
  let A be OSSubset of OU0;
  reconsider X = the Sorts of OU0 as Element of SubSort(A) by MSUALG_2:38;
  the Sorts of OU0 is OrderSortedSet of S1 by OSALG_1:17;
  then
  X in { x where x is Element of SubSort(A): x is OrderSortedSet of S1};
  hence thesis;
end;
