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 Th13:
  for U1,U2 be non-empty MSAlgebra over S
  for F be ManySortedFunction of U1,U2
  for x be Element of Args(o,U1) for n be set st
  n in dom the_arity_of o holds (F#x).n = F.((the_arity_of o)/.n).(x.n)
proof
  let U1,U2 be non-empty MSAlgebra over S;
  let F be ManySortedFunction of U1,U2;
  let x be Element of Args(o,U1);
  let n be set such that
A1: n in dom the_arity_of o;
  dom F = the carrier of S by PARTFUN1:def 2; then
a1: rng the_arity_of o c= dom F; then
A2: n in dom (F*the_arity_of o) by A1,RELAT_1:27;
B2: dom x = dom the_arity_of o by MSUALG_6:2;
    dom (F*the_arity_of o) = dom the_arity_of o by a1,RELAT_1:27; then
    dom ((F*the_arity_of o)..x) = (dom the_arity_of o) /\ dom x
      by PRALG_1:def 19
   .= dom the_arity_of o by B2; then
a2: n in dom ((F*the_arity_of o)..x) by A1;
A3: x in product doms (F*the_arity_of o) by Th12;
  thus (F#x).n = ((Frege(F*the_arity_of o)).x).n by MSUALG_3:def 5
    .= ((F*the_arity_of o)..x).n by A3,PRALG_2:def 2
    .= ((F*the_arity_of o).n).(x.n) by a2,PRALG_1:def 19
    .= (F.((the_arity_of o).n)).(x.n) by A2,FUNCT_1:12
    .= F.((the_arity_of o)/.n).(x.n) by A1,PARTFUN1:def 6;
end;
