
theorem LM28:
  for X being RealNormSpace-Sequence,
      x being Point of product X
  st ( for i be Element of dom X holds ||.x.i.|| <= 1 )
  holds 0 <= NrProduct x & NrProduct x <= 1
  proof
    let X be RealNormSpace-Sequence,
        x be Point of product X;
    assume
    A1: for i be Element of dom X holds ||.x.i.|| <= 1;
    consider F be FinSequence of REAL such that
    A2: dom F = dom X
      & ( for i be Element of dom X holds F.i = ||.x.i.|| )
      & NrProduct x = Product F by DefNrPro;
    for i be Element of dom F holds 0 <= F.i & F.i <= 1
    proof
      let i be Element of dom F;
      reconsider j = i as Element of dom X by A2;
      A3: F.j = ||.x.j.|| by A2;
      thus 0 <= F.i by A3;
      thus F.i <= 1 by A1,A3;
    end;
    hence 0 <= NrProduct x & NrProduct x <= 1 by A2,LM281;
  end;
