
theorem
  for X be RealNormSpace-Sequence,
      x be Point of product X
  holds
    ( (ex i be Element of dom X st x.i = 0.(X.i)) iff NrProduct x = 0 )
  & ( not (ex i be Element of dom X st x.i = 0.(X.i)) implies 0 < NrProduct x )
  proof
    let X be RealNormSpace-Sequence,
        x be Point of product X;
    consider Nx be FinSequence of REAL such that
    A1: dom Nx = dom X & (for i be Element of dom X holds Nx.i = ||.x.i.||) and
    A2: NrProduct x = Product Nx by DefNrPro;
    thus ( ex i be Element of dom X st x.i=0.(X.i) ) iff NrProduct x = 0
    proof
      hereby
        assume ex i be Element of dom X st x.i=0.(X.i); then
        consider i be Element of dom X such that
        A3: x.i = 0.(X.i);
        Nx.i = ||.0.(X.i).|| by A1,A3
        .= 0;
        hence NrProduct x = 0 by A1,A2,RVSUM_1:103;
      end;
      assume NrProduct x = 0; then
      consider k be Nat such that
      A5: k in dom Nx & Nx.k = 0 by A2,RVSUM_1:103;
      reconsider i = k as Element of dom X by A1,A5;
      ||.x.i.|| = 0 by A1,A5; then
      x.i = 0.(X.i) by NORMSP_0:def 5;
      hence thesis;
    end;
    thus not (ex i be Element of dom X st x.i = 0.(X.i))
      implies 0 < NrProduct x
    proof
      assume
      A7: not ex i be Element of dom X st x.i=0.(X.i);
      for k being Element of NAT st k in dom Nx holds Nx.k > 0
      proof
        let k be Element of NAT;
        assume k in dom Nx; then
        reconsider i = k as Element of dom X by A1;
        A9: Nx.i = ||.x.i.|| by A1;
        x.i <> 0.(X.i) by A7; then
        Nx.k <> 0 by A9,NORMSP_0:def 5;
        hence Nx.k > 0 by A9;
      end;
      hence 0 < NrProduct x by A2,NAT_4:42;
    end;
  end;
