reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th43:
  for f,g being non empty Element of HFuncs NAT,
      F being with_the_same_arity FinSequence of HFuncs NAT
  st g = f*<:F:> holds arity g = arity F
proof
  let f, g be non empty Element of HFuncs NAT, F be with_the_same_arity
  FinSequence of HFuncs NAT;
  assume g = f*<:F:>;
  then
A1: dom g c= (arity F)-tuples_on NAT by Th41;
  consider x being object such that
A2: x in dom g by XBOOLE_0:def 1;
  reconsider x as Element of (arity F)-tuples_on NAT by A1,A2;
  len x = arity F by CARD_1:def 7;
  hence thesis by A2,MARGREL1:def 25;
end;
