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 Th15:
  for x be Element of Args(o,product A) for n be set st n in dom (
  the_arity_of o) holds x.n in product Carrier(A,(the_arity_of o)/.n)
proof
  let x be Element of Args(o,product A);
  let n be set such that
A1: n in dom (the_arity_of o);
  dom (SORTS A) = the carrier of S by PARTFUN1:def 2;
  then rng (the_arity_of o) c= dom (SORTS A);
  then
A2: n in dom ((SORTS A)*(the_arity_of o)) by A1,RELAT_1:27;
  x in Args(o,product A);
  then x in product ((the Sorts of (product A))*(the_arity_of o)) by PRALG_2:3;
  then x.n in ((SORTS A)*(the_arity_of o)).n by A2,CARD_3:9;
  then x.n in (SORTS A).((the_arity_of o).n) by A2,FUNCT_1:12;
  then x.n in (SORTS A).((the_arity_of o)/.n) by A1,PARTFUN1:def 6;
  hence thesis by PRALG_2:def 10;
end;
