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 Th19:
  for I be set, S be non void non empty ManySortedSign, A be
  MSAlgebra-Family of I,S, o be OperSymbol of S for x be Element of Args(o,
product A) holds Den(o,product A).x in product Carrier(A,the_result_sort_of o)
proof
  let I be set, S be non void non empty ManySortedSign, A be MSAlgebra-Family
  of I,S, o be OperSymbol of S;
  let x be Element of Args(o,product A);
  Result(o,product A) = (SORTS A).(the_result_sort_of o) by PRALG_2:3
    .= product Carrier(A,the_result_sort_of o) by PRALG_2:def 10;
  hence thesis;
end;
