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 Th19:
  for A,B be OSSubset of OU0 holds B in OSSubSort(A) iff B is
  opers_closed & OSConstants(OU0) c= B & A c= B
proof
  let A, B be OSSubset of OU0;
  thus B in OSSubSort(A) implies B is opers_closed & OSConstants(OU0) c= B & A
  c= B
  proof
    assume B in OSSubSort(A);
    then
A1: ex B1 being Element of SubSort(A) st B1 = B & B1 is OrderSortedSet of S1;
    then Constants(OU0) c= B by MSUALG_2:13;
    hence thesis by A1,Th11,MSUALG_2:13;
  end;
  assume that
A2: B is opers_closed and
A3: OSConstants(OU0) c= B and
A4: A c= B;
  Constants(OU0) c= OSConstants(OU0) by Th10;
  then Constants(OU0) c= B by A3,PBOOLE:13;
  then
A5: B in SubSort(A) by A2,A4,MSUALG_2:13;
  B is OrderSortedSet of S1 by Def2;
  hence thesis by A5;
end;
