
theorem LM32:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace,
      g be MultilinearOperator of X,Y,
      t be Point of product X
  st ex i be Element of dom X st t.i = 0.(X.i)
  holds g.t = 0.Y
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace,
        g be MultilinearOperator of X,Y,
        t be Point of product X;
    given i be Element of dom X such that
    A2: t.i = 0.(X.i);
    A6: (g * reproj(i,t)).( 0.(X.i) )
     = g.(reproj(i,t).( 0.(X.i) )) by FUNCT_2:15
    .= g.t by A2,Th4X;
    g * reproj(i,t) is LinearOperator of X.i,Y by Def3;
    hence g.t = 0.Y by A6,LOPBAN_7:3;
  end;
