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 Th29:
  for A be OSSubset of OU0 holds OSMSubSort(A) is opers_closed & A
  c= OSMSubSort(A)
proof
  let A be OSSubset of OU0;
  thus OSMSubSort(A) is opers_closed
  proof
    let o1 be Element of the carrier' of S1;
    reconsider o = o1 as OperSymbol of S1;
    rng ((Den(o,OU0))|(((OSMSubSort A)# * (the Arity of S1)).o)) c= ((
    OSMSubSort A) * (the ResultSort of S1)).o by Th28;
    hence thesis;
  end;
  A c= OSConstants(OU0) (\/) A & OSConstants(OU0) (\/) A c= OSMSubSort(A)
    by Th24,PBOOLE:14;
  hence thesis by PBOOLE:13;
end;
