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 Th23:
  for A being OSSubset of OU0, s being SortSymbol of S1 holds (the
  Sorts of OU0).s in OSSubSort(A,s)
proof
  let A be OSSubset of OU0, s be SortSymbol of S1;
  the Sorts of OU0 is ManySortedSubset of the Sorts of OU0 & the Sorts of
  OU0 is OrderSortedSet of S1 by OSALG_1:17,PBOOLE:def 18;
  then reconsider B = the Sorts of OU0 as OSSubset of OU0 by Def2;
  the Sorts of OU0 in OSSubSort(A) by Th17;
  then B.s in OSSubSort(A,s) by Def10;
  hence thesis;
end;
