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 Th9:
  for o be OperSymbol of S st the_arity_of o = {} holds (const (o,
  product A)).i = const (o,A.i)
proof

  let o be OperSymbol of S such that
A1: the_arity_of o = {};
  set f = (commute (OPER A)).o, C = union the set of all
 Result(o,A.i9) where i9 is Element
  of I;
A2: f in Funcs(I,Funcs({{}},C)) by A1,Th7;
  (OPS A).o = (IFEQ(the_arity_of o,{},commute(A?.o),Commute Frege(A?.o)))
  by PRALG_2:def 13
    .= commute(A?.o) by A1,FUNCOP_1:def 8;
  then
A3: const(o,product A) = (commute f).{} by MSUALG_1:def 6;
A4: {} in {{}} by TARSKI:def 1;
  const (o,A.i) = ((A?.o).i).{} by PRALG_2:7
    .= const(o,product A).i by A2,A3,A4,FUNCT_6:56;
  hence thesis;
end;
