reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th21:
  for Ft be FinSequence of TOP-REAL n,
      Fr be FinSequence of REAL-NS n
    st Ft = Fr
  holds Sum Ft = Sum Fr
  proof
    let F be FinSequence of TOP-REAL n,
       Fv be FinSequence of REAL-NS n;

    assume
    A1: F = Fv;
    set T = TOP-REAL n;
    set V = REAL-NS n;

    consider f be sequence of the carrier of T such that
    A2: Sum F = f.(len F) and
    A3: f.0 = 0.T and
    A4: for j be Nat
        for v be Element of T
          st j < len F & v = F.(j + 1)
        holds f.(j + 1) = f.j + v by RLVECT_1:def 12;

    consider fv be sequence of the carrier of V such that
    A5: Sum Fv = fv.(len Fv) and
    A6: fv.0 = 0.V and
    A7: for j be Nat
        for v be Element of V
          st j < len Fv & v = Fv.(j + 1)
        holds fv.(j + 1) = fv.j + v by RLVECT_1:def 12;

    defpred S1[ Nat] means
    ($1 <= len F implies f.$1 = fv.$1);

    A8: for i be Nat st S1[i] holds S1[i + 1]
    proof
      let i be Nat;
      assume
      A9: S1[i];
      assume
      A10: i + 1 <= len F; then
      A11: i + 1 in dom F by NAT_1:11,FINSEQ_3:25;
      then F.(i + 1) = F /. (i + 1) by PARTFUN1:def 6;
      then
      A12: f.(i + 1) = f.i + F /. (i + 1) by A4, A10,NAT_1:13;
      A13: Fv /. (i + 1) = Fv.(i + 1) by A1,A11,PARTFUN1:def 6;
      then Fv /. (i + 1) = F /. (i + 1) by A1,A11,PARTFUN1:def 6;
      hence f.(i + 1)
       = fv.i + Fv /. (i + 1) by A9,A10,A12,Th7,NAT_1:13
      .= fv.(i + 1) by A1,A7,A10,A13,NAT_1:13;
    end;
    A14: S1[ 0 ] by A3,A6,Th6;
    for n be Nat holds S1[n] from NAT_1:sch 2(A14, A8);
    hence Sum F = Sum Fv by A1,A2,A5;
  end;
