
theorem
  for X being RealLinearSpace-Sequence,
      Y being RealLinearSpace,
      g be MultilinearOperator of X,Y,
      t be Point of product X,
      s be Element of product carr X
  st s = t
   & ex i be Element of dom X st s.i = 0.(X.i)
  holds g.t = 0.Y
  proof
    let X be RealLinearSpace-Sequence,
        Y be RealLinearSpace,
        g be MultilinearOperator of X,Y,
        t be Point of product X,
        s be Element of product carr X;
    assume that
    A1: s = t and
    A2: ex i be Element of dom X st s.i = 0.(X.i);
    consider i be Element of dom X such that
    A3: s.i = 0.(X.i) by A2;
    A7: ( g * reproj(i,t) ). ( 0.(X.i) )
      = g.(reproj (i,t).( 0.(X.i) ) ) by FUNCT_2:15
     .= g.t by A1,A3,Th4X;
    g * reproj(i,t) is LinearOperator of X.i,Y by Def3;
    hence g.t = 0.Y by A7,LOPBAN73;
  end;
