reserve x,y,z for set;

theorem Th21:
  for S being non void Signature for Y being non-empty
ManySortedSet of the carrier of S for X being ManySortedSet of the carrier of S
  holds S-Terms(X,Y) is opers_closed
proof
  let S be non void Signature;
  let Y be non-empty ManySortedSet of the carrier of S;
  let X be ManySortedSet of the carrier of S;
  for o being OperSymbol of S for p being ArgumentSeq of Sym(o,Y) st rng p
  c= Union (S-Terms(X,Y)) holds Sym(o,Y)-tree p in (S-Terms(X,Y)).
  the_result_sort_of o by Th19;
  hence thesis by Th20;
end;
