reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th36:
  for e being FinSequence of REAL for f being FinSequence of
  F_Real st e = f holds Sum e = Sum f
proof
  let e be FinSequence of REAL;
  let f be FinSequence of F_Real such that
A1: e = f;
  consider e1 being sequence of REAL such that
A2: e1.0 = 0 and
A3: for i be Nat st i < len e holds e1.(i+1) = e1.i+e.(i+1) and
A4: Sum e = e1.len e by Th7;
  consider f1 being sequence of the carrier of F_Real such that
A5: Sum f = f1.len f and
A6: f1.0 = 0.F_Real and
A7: for j being Nat
    for v being Element of F_Real st j < len f & v = f.(j + 1)
  holds f1.(j + 1) = f1.j + v by RLVECT_1:def 12;
  for n holds n <= len e implies e1.n = f1.n
  proof
    defpred p[Nat] means $1 <= len e implies e1.$1 = f1.$1;
    let n;
A8: now
      let k be Nat such that
A9:   p[k];
      now
        reconsider k9=k as Element of NAT by ORDINAL1:def 12;
        e.(k+1) in REAL by XREAL_0:def 1;
        then reconsider a = e.(k+1) as Element of F_Real by VECTSP_1:def 5;
        assume k+1 <= len e;
        then
A10:    k < len e by NAT_1:13;
        then e1.(k+1) = f1.k9 + a by A3,A9
          .= f1.(k+1) by A1,A7,A10;
        hence e1.(k+1) = f1.(k+1);
      end;
      hence p[k+1];
    end;
A11: p[0] by A2,A6,STRUCT_0:def 6,VECTSP_1:def 5;
    for n be Nat holds p[n] from NAT_1:sch 2(A11,A8);
    hence thesis;
  end;
  hence thesis by A1,A4,A5;
end;
