reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;
reserve FSeq for FinSequence of Sigma;

theorem Th63:
  len RFin >= 1 implies ex f being Real_Sequence st f.1 = RFin.1 &
  (for n st 0 <> n & n < len RFin holds f.(n+1) = f.n+RFin.(n+1)) &
  Sum(RFin) = f.(len RFin)
proof
  assume
A0: len RFin >= 1;
  reconsider RFin as FinSequence of REAL by FINSEQ_1:106;
  consider f be sequence of REAL such that
A1: f.1 = RFin.1 and
A2: for n being Nat st 0 <> n & n < len RFin holds f.(n+1) =
  addreal.(f.n,RFin.(n+1)) and
A3: addreal "**" RFin = f.(len RFin) by A0,FINSOP_1:1;
  take f;
  for n st 0 <> n & n < len RFin holds f.(n+1) = f.n+ RFin.(n+1)
  proof
    let n;
    reconsider n1=n as Element of NAT by ORDINAL1:def 12;
    assume that
A4: 0 <> n and
A5: n < len RFin;
    thus f.(n+1) = addreal.(f.n1,RFin.(n1+1)) by A2,A4,A5
      .= f.n+ RFin.(n+1) by BINOP_2:def 9;
  end;
  hence thesis by A1,A3,RVSUM_1:def 12;
end;
