reserve I for non empty set,
  J for ManySortedSet of I,
  S for non void non empty ManySortedSign,
  i for Element of I,
  c for set,
  A for MSAlgebra-Family of I,S,
  EqR for Equivalence_Relation of I,
  U0,U1,U2 for MSAlgebra over S,
  s for SortSymbol of S,
  o for OperSymbol of S,
  f for Function;

theorem Th18:
  for y be Element of Args(o,product A) st the_arity_of o <> {}
  holds y in dom (Commute Frege(A?.o))
proof
  let y be Element of Args(o,product A);
  set D = union the set of all
 (the Sorts of A.ii).ss where ii is Element of I,ss is
  Element of (the carrier of S) ;
  assume
A1: the_arity_of o <> {};
  then (commute y) in product doms (A?.o) by Th17;
  then
A2: (commute y) in dom (Frege(A?.o)) by PARTFUN1:def 2;
  y in Funcs (dom (the_arity_of o),Funcs (I,D)) by Th14;
  then y = commute commute y by A1,FUNCT_6:57;
  hence thesis by A2,PRALG_2:def 1;
end;
