
theorem
  for X be RealNormSpace holds
  0.(product <*X*>) = (IsoCPNrSP X).(0.X)
  proof
    let X be RealNormSpace;
    consider I be Function of X, product <*X*> such that
    I is one-to-one onto and
    A1: for x be Point of X holds I.x = <*x*> and
    A2: for v,w be Point of X holds I.(v+w) = I.v + I.w and
    A3: for v be Point of X, r be Element of REAL
        holds I.(r*v) = r*(I.v) and
    A4: 0. product <*X*> = I.(0.X) and
    for v be Point of X holds ||.I.v.|| = ||.v.|| by PRVECT_3:16;

    now
      let v be Point of X, r be Real;
      reconsider r0 = r as Element of REAL by XREAL_0:def 1;

      thus
      I.(r*v)
       = r0 * I.v by A3
      .= r * I.v;
    end; then
    reconsider I as LinearOperator of X, product <*X*>
      by A2,LOPBAN_1:def 5,VECTSP_1:def 20;
    for a be Element of X holds I.a = (IsoCPNrSP X).a by A1,Def2;
    hence thesis by A4;
  end;
