
theorem
  for X being RealNormSpace-Sequence,
      s being Element of product X
  ex F be FinSequence of REAL
  st dom F = dom X
   & for i be Element of dom X
  holds F.i=||.s.i.||
  proof
    let X be RealNormSpace-Sequence,
        s be Element of product X;
    defpred P1[object, object] means
    ex i be Element of dom X
    st $1 = i & $2 = ||.s.i.||;
    A5: for n being Nat st n in Seg len X holds
    ex d being Element of REAL st P1[n,d]
    proof
      let n be Nat;
      assume n in Seg len X; then
      reconsider i = n as Element of dom X by FINSEQ_1:def 3;
      reconsider d = ||.s.i.|| as Element of REAL;
      take d;
      thus P1[n,d];
    end;
    consider F being FinSequence of REAL such that
    A6: len F = len X
      & for n being Nat st n in Seg len X holds
        P1[n,F /. n] from FINSEQ_4:sch 1(A5);
    take F;
    thus
    A7: dom F = dom X by A6,FINSEQ_3:29;
    thus for i be Element of dom X holds F.i=||.s.i.||
    proof
      let i be Element of dom X;
      i in dom X; then
      A8: i in Seg len X by FINSEQ_1:def 3;
      reconsider n = i as Nat;
      consider j be Element of dom X such that
      A9: n = j & F/.n = ||.s.j.|| by A6,A8;
      thus F.i = ||.s.i.|| by A7,A9,PARTFUN1:def 6;
    end;
  end;
