reserve X,G for RealNormSpace-Sequence,
          Y for RealNormSpace;
reserve f for MultilinearOperator of X,Y;

theorem ZERXI:
  for z be Element of product X st z = 0.product X
  holds for i be Element of dom X holds z.i = 0.(X.i)
  proof
    let z be Element of product X;
    assume
    A1: z = 0.product X;
    let i be Element of dom X;
    A2: product X = NORMSTR(# (product (carr X)),(zeros X),
    [:(addop X):],[:(multop X):],(productnorm X) #) by PRVECT_2:6;
    reconsider j = i as Element of dom carr X by DCARXX;
    (zeros X).j = 0.(X.j) by PRVECT_1:def 14;
    hence z.i = 0.(X.i) by A1,A2;
  end;
