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 Th8:
  the_arity_of o = {} implies const(o,product A) in Funcs(I, union
  the set of all  Result(o,A.i9) where i9 is Element of I)
proof
  set g = (commute (OPER A)).o;
  set C = union the set of all  Result(o,A.i9) where i9 is Element of I;
  assume
A1: the_arity_of o = {};
  then
A2: g in Funcs(I,Funcs({{}},C)) by Th7;
  reconsider g as Function;
  (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 g).{} by MSUALG_1:def 6;
  commute g in Funcs({{}},Funcs(I,C)) by A2,FUNCT_6:55;
  then consider g1 be Function such that
A4: g1 = commute g and
A5: dom g1 = {{}} and
A6: rng g1 c= Funcs(I,C) by FUNCT_2:def 2;
  {} in {{}} by TARSKI:def 1;
  then g1.{} in rng g1 by A5,FUNCT_1:def 3;
  hence thesis by A3,A4,A6;
end;
