theorem Th71:
  for f being Element of P, F being with_the_same_arity FinSequence of P
   st arity f = len F holds f*<:F:> in P
proof
  let f be Element of P, F being with_the_same_arity FinSequence of P;
  assume
A1: arity f = len F;
A2: rng F c= P by FINSEQ_1:def 4;
  per cases;
  suppose
    f is empty;
    then f*<:F:> = {};
    hence thesis by Th70;
  end;
  suppose
    f is non empty;
    then reconsider f9 = f as non empty Element of HFuncs NAT;
A3: P* c= (HFuncs NAT)* by FINSEQ_1:62;
    F in P* by FINSEQ_1:def 11;
    then reconsider F9 = F as with_the_same_arity Element of (HFuncs NAT)*
    by A3;
    P is composition_closed by Def14;
    then f9*<:F9:> in P by A1,A2;
    hence thesis;
  end;
end;
