
theorem Th19:
  for X be RealNormSpace,
      x be Point of product <*X*>
  holds NrProduct x = ||.x.||
  proof
    let X be RealNormSpace,
        x be Point of product <*X*>;

    A1: dom <*X*> = {1} by FINSEQ_1:2,38;

    consider x1 be Point of X such that
    A3: x = <*x1*> by Th12;

    rng <*||.x1.||*> c= REAL; then
    reconsider Nx = <*||.x1.||*> as FinSequence of REAL by FINSEQ_1:def 4;

    A4: dom Nx = {1} by FINSEQ_1:2,38;

    A5: now
      let i be Element of dom <*X*>;
      A6: i = 1 by A1,TARSKI:def 1;
      hence
      Nx.i = ||.x1.||
          .= ||.x.i.|| by A3,A6;
    end;

    Product Nx = ||.x1.||;
    hence thesis by A1,A3,A4,A5,Th12,LOPBAN10:def 9;
  end;
