
theorem LmSign1C:
  for F being FinSequence of F_Real
  st for i being Nat st i in dom F holds F.i in F_Rat
  holds Sum F in F_Rat
  proof
    defpred P[Nat] means
    for F being FinSequence of F_Real
    st len F = $1 &
    for i being Nat st i in dom F holds F.i in F_Rat
    holds Sum F in F_Rat;
    P1: P[0]
    proof
      let F be FinSequence of F_Real;
      assume AS1: len F = 0 &
      for i being Nat st i in dom F holds F.i in F_Rat;
      F = <*> the carrier of F_Real by AS1;
      then Sum F = 0.F_Real by RLVECT_1:43
      .= 0;
      hence Sum F in F_Rat;
    end;
    P2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume AS1: P[n];
      let F be FinSequence of F_Real;
      assume AS2: len F = n+1 &
      for i being Nat st i in dom F holds F.i in F_Rat;
      reconsider F0 = F| n as FinSequence of F_Real;
      n+1 in Seg (n+1) by FINSEQ_1:4; then
      A70: n+1 in dom F by AS2,FINSEQ_1:def 3;
      then F.(n+1) in rng F by FUNCT_1:3;
      then reconsider af = F.(n+1) as Element of F_Real;
      P1: len F0 = n by FINSEQ_1:59,AS2,NAT_1:11; then
      P4: dom F0 = Seg n by FINSEQ_1:def 3;
      len F = (len F0) + 1 by AS2,FINSEQ_1:59,NAT_1:11; then
      P3: Sum F = Sum F0 + af by AS2,P4,RLVECT_1:38;
      for i being Nat st i in dom F0 holds F0.i in F_Rat
      proof
        let i be Nat;
        assume P40: i in dom F0;
        dom F = Seg (n+1) by AS2,FINSEQ_1:def 3;
        then dom F0 c= dom F by P4,FINSEQ_1:5,NAT_1:11;
        then F.i in F_Rat by AS2,P40;
        hence thesis by P40,FUNCT_1:47;
      end;
      then Sum F0 in F_Rat by P1,AS1;
      then reconsider i1 = Sum F0 as Element of F_Rat;
      F.(n+1) in F_Rat by A70,AS2;
      then reconsider i2 = af as Element of F_Rat;
      Sum F = i1 + i2 by P3;
      hence Sum F in F_Rat;
    end;
    X1: for n being Nat holds P[n] from NAT_1:sch 2(P1,P2);
    let F be FinSequence of F_Real;
    assume X2: for i being Nat st i in dom F holds F.i in F_Rat;
    len F is Nat;
    hence Sum F in F_Rat by X1,X2;
  end;
