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

theorem FXZER:
  f.(0.product X) = 0.Y
  proof
    A1: product X = NORMSTR(# (product (carr X)),(zeros X),
    [:(addop X):],[:(multop X):],(productnorm X) #) by PRVECT_2:6;
    reconsider z = 0.product X as Element of product carr X by A1;
    set i = the Element of dom X;
    z.i = 0.(X.i) by ZERXI;
    hence f.(0.product X) = 0.Y by LOPBAN10:36;
  end;
