 reserve RNS1,RNS2 for RealLinearSpace;

theorem Th13:
the RLSStruct of RNS1 = the RLSStruct of RNS2
implies
  for Ft be FinSequence of RNS1,
      FFr be FinSequence of RNS2 st Ft = FFr
  holds Sum Ft = Sum FFr
proof
assume A1: the RLSStruct of RNS1 = the RLSStruct of RNS2;
let F be FinSequence of RNS1,
    Fv be FinSequence of RNS2;
assume A2: F = Fv;
set T = RNS1;
set V = RNS2;
consider f being sequence of the carrier of T such that
A3: Sum F = f.(len F) and
A4: f.0 = 0.T and
A5: for j being Nat
    for v being 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 being sequence of the carrier of V such that
A6: Sum Fv = fv.(len Fv) and
A7: fv.0 = 0.V and
A8: for j being Nat
    for v being 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;
A9: for i being Nat st S1[i] holds S1[i+1]
proof
  let i be Nat;
  assume A10: S1[i];
  assume A11: i+1 <= len F; then
  A12: i+1 in dom F by NAT_1:11, FINSEQ_3:25; then
  F.(i+1) = F/.(i+1) by PARTFUN1:def 6; then
  A13: f.(i+1) = (f.i) + F/.(i+1) by A5, A11, NAT_1:13;
  A14: Fv/.(i+1) = Fv.(i+1) by A2, A12, PARTFUN1:def 6;
  thus f.(i+1) = fv.i + Fv/.(i+1) by A10, A11, A13, NAT_1:13, A1, A2
              .= fv.(i+1) by A2, A8, A11, NAT_1:13, A14;
end;
A15: S1[ 0 ] by A4, A7, A1;
for n being Nat holds S1[n] from NAT_1:sch 2(A15, A9);
hence Sum F = Sum Fv by A2, A3, A6;
end;
