theorem
  fire(<*>N) = id Funcs(P, NAT)
proof
  consider F being Function-yielding FinSequence such that
A1: fire(<*>N) = compose(F, Funcs(P, NAT)) and
A2: len F = len <*>N and
  for i being Element of NAT st i in dom <*>N holds
  F.i = fire ((<*>N)/.i qua Element of N) by Def10;
  F = {} by A2;
  hence thesis by A1,FUNCT_7:39;
end;
