
theorem
  for c be Complex holds 0 (#) (NAT --> c) = NAT --> 0
  proof
    let c be Complex;
    A1: dom (0 (#) (NAT --> c)) = dom (NAT --> c) by VALUED_1:def 5
    .= dom (NAT --> 0);
    for k be object st k in dom (NAT --> 0) holds
      (NAT --> 0).k = (0 (#) (NAT --> c)).k
    proof
      let k be object such that
      B1: k in dom (NAT --> 0);
      reconsider k as Nat by B1;
      (NAT --> 0).k = 0 * (NAT --> c).k
      .= (0 (#) (NAT --> c)).k by A1,B1,VALUED_1:def 5;
      hence thesis;
    end;
    hence thesis by A1;
  end;
