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 Th21:
  for A being OSSubset of OU0, s1,s2 being SortSymbol of S1 holds
  s1 <= s2 implies OSSubSort(A,s2) is_coarser_than OSSubSort(A,s1)
proof
  let A be OSSubset of OU0, s1,s2 be SortSymbol of S1;
  assume
A1: s1 <= s2;
  for Y being set st Y in OSSubSort(A,s2) ex X being set st X in OSSubSort
  (A,s1) & X c= Y
  proof
    let x be set;
    assume x in OSSubSort(A,s2);
    then consider B being OSSubset of OU0 such that
A2: B in OSSubSort(A) & x = B.s2 by Def10;
    take B.s1;
    B is OrderSortedSet of S1 by Def2;
    hence thesis by A1,A2,Def10,OSALG_1:def 16;
  end;
  hence thesis by SETFAM_1:def 3;
end;
