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
  the_arity_of o = {} & dom f = I & (for i be Element of I holds f.i =
  const(o,A.i)) implies f = const(o,product A)
proof
  assume that
A1: the_arity_of o = {} and
A2: dom f = I and
A3: for i be Element of I holds f.i = const(o,A.i);
A4: now
    let a be object;
    assume a in I;
    then reconsider a9 = a as Element of I;
    thus f.a = const(o,A.a9) by A3
      .= (const(o,product A)).a by A1,Th9;
  end;
  set C = union the set of all  Result(o,A.i9) where i9 is Element of I;
  const(o,product A) in Funcs(I,C) by A1,Th8;
  then ex g2 be Function st g2 = const(o,product A) & dom g2 = I & rng g2 c= C
  by FUNCT_2:def 2;
  hence thesis by A2,A4;
end;
