theorem Th70:
  {} in P
proof
  reconsider F = {} as with_the_same_arity Element of (HFuncs NAT)* by
FINSEQ_1:49;
  set f = 0 const 0;
A1: rng {} c= P;
A2: arity f = 0;
A3: P is composition_closed by Def14;
  f in P by Th68;
  then f*<:F:> in P by A2,A1,A3,CARD_1:27;
  hence thesis by FUNCT_6:40;
end;
