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 Th20:
  for B be OSSubset of OU0 holds B in OSSubSort(OU0) iff B is opers_closed
proof
  let B be OSSubset of OU0;
A1: B in SubSort(OU0) iff B is opers_closed by MSUALG_2:14;
  thus B in OSSubSort(OU0) implies B is opers_closed
  proof
    assume B in OSSubSort(OU0);
    then
    ex B1 being Element of SubSort(OU0) st B1 = B & B1 is OrderSortedSet of S1;
    hence thesis by MSUALG_2:14;
  end;
  assume
A2: B is opers_closed;
  B is OrderSortedSet of S1 by Def2;
  hence thesis by A1,A2;
end;
