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 Th14:
  for x be Element of Args(o,product A) holds x in Funcs (dom (
the_arity_of o),Funcs (I,union the set of all
 (the Sorts of A.i9).s9 where i9 is Element of
  I,s9 is Element of (the carrier of S) ))
proof
  let x be Element of Args(o,product A);
  set C = union the set of all
 (the Sorts of A.i9).s9 where i9 is Element of I, s9 is
  Element of (the carrier of S) ;
  consider x1 be Function such that
A1: x1 = x;
  x in Args(o,product A);
  then
A2: x in product ((the Sorts of (product A))*(the_arity_of o)) by PRALG_2:3;
  dom (SORTS A) = the carrier of S by PARTFUN1:def 2;
  then
A3: rng (the_arity_of o) c= dom (SORTS A);
  now
    let c be object;
    assume c in rng x1;
    then consider n be object such that
A4: n in dom x1 and
A5: x1.n = c by FUNCT_1:def 3;
A6: n in dom ((SORTS A)*(the_arity_of o)) by A2,A1,A4,CARD_3:9;
    then n in dom (the_arity_of o) by A3,RELAT_1:27;
    then (the_arity_of o).n in rng (the_arity_of o) by FUNCT_1:def 3;
    then reconsider s1 = (the_arity_of o).n as Element of (the carrier of S);
    x1.n in ((SORTS A)*(the_arity_of o)).n by A2,A1,A6,CARD_3:9;
    then x1.n in (SORTS A).s1 by A6,FUNCT_1:12;
    then x1.n in product Carrier(A,s1) by PRALG_2:def 10;
    then consider g be Function such that
A7: g = x1.n and
A8: dom g = dom Carrier(A,s1) and
A9: for i9 be object st i9 in dom (Carrier(A,s1)) holds g.i9 in (Carrier
    (A,s1)).i9 by CARD_3:def 5;
    now
      let c1 be object;
      assume c1 in rng g;
      then consider i1 be object such that
A10:  i1 in dom g and
A11:  g.i1 = c1 by FUNCT_1:def 3;
      reconsider i1 as Element of I by A8,A10;
      ex U0 being MSAlgebra over S st U0 = A.i1 & (Carrier(A,s1 )).i1 = (
      the Sorts of U0).s1 by PRALG_2:def 9;
      then
A12:  g.i1 in (the Sorts of A.i1).s1 by A8,A9,A10;
      (the Sorts of A.i1).s1 in the set of all
(the Sorts of A.i9).s9 where i9 is
      Element of I, s9 is Element of (the carrier of S) ;
      hence c1 in C by A11,A12,TARSKI:def 4;
    end;
    then
A13: rng g c= C;
    dom g = I by A8,PARTFUN1:def 2;
    hence c in Funcs(I,C) by A5,A7,A13,FUNCT_2:def 2;
  end;
  then
A14: rng x1 c= Funcs(I,C);
  dom x = dom ((SORTS A)*(the_arity_of o)) by A2,CARD_3:9
    .= dom (the_arity_of o) by A3,RELAT_1:27;
  hence thesis by A1,A14,FUNCT_2:def 2;
end;
