
theorem LM33:
  for X being RealNormSpace-Sequence,
      s be Point of product X,
      a be FinSequence of REAL
  ex s1 be Point of product X
  st for i be Element of dom X holds s1.i = (a/.i) * s.i
  proof
    let X be RealNormSpace-Sequence,
        x be Point of product X,
        a be FinSequence of REAL;
    A4: dom carr X = dom X by LemmaX;
    defpred P1[object, object] means
    ex i be Element of dom X
    st $1 = i & $2=a/.i * x.i;
    A5: for n being Nat st n in Seg len X holds
        ex d being object 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 = a/.i * x.i as Element of X.i;
      take d;
      thus P1[n,d];
    end;
    consider F being FinSequence such that
    A6: dom F = Seg len X
      & for n being Nat st n in Seg len X holds
        P1[n,F . n] from FINSEQ_1:sch 1(A5);
    A7: dom F = dom carr X by A4,A6,FINSEQ_1:def 3;
    for y being object st y in dom carr X holds F.y in (carr X).y
    proof
      let y be object;
      assume
      A8: y in dom carr X; then
      reconsider n = y as Nat;
      consider i be Element of dom X such that
      A9: n = i & F.n = a/.i * x.i by A6,A7,A8;
      F.n in the carrier of (X.i) by A9;
      hence F.y in (carr X).y by A9,PRVECT_1:def 11;
    end; then
    reconsider F as Element of product carr X by A7,CARD_3:def 5;
    reconsider F as Element of product X by EXTh10;
    take F;
    thus for i be Element of dom X holds F.i = a/.i * x.i
    proof
      let i be Element of dom X;
      i in dom X; then
      A10: i in Seg len X by FINSEQ_1:def 3;
      set n = i;
      consider j be Element of dom X such that
      A11: n = j & F.n = a/.j * x.j by A6,A10;
      thus F.i = a/.i* x.i by A11;
    end;
  end;
