reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th2:
  for Z being complex-valued FinSequence, E being FinSequence of F_Complex st
    E=Z holds Sum Z = Sum E
proof
  let Z be complex-valued FinSequence,E be FinSequence of FC such that
A1: E=Z;
  consider f be sequence of FC such that
A2: Sum E = f.(len E) & f.0 = 0.FC and
A3: for j being Nat,v be Element of FC
    st j < len E & v = E.(j + 1) holds f.(j + 1) = f.j + v
    by RLVECT_1:def 12;
  reconsider E1=E as FinSequence of FC;
  defpred P[Nat] means $1 <= len Z implies Sum (Z|$1) = f.$1;
A4: P[0] by A2,RVSUM_2:29,COMPLFLD:7;
A5: P[n] implies P[n+1]
  proof set n1=n+1;
    assume
A6:   P[n] & n1 <= len Z;
A7: n1 in dom Z by A6,NAT_1:11, FINSEQ_3:25;
    then
A8: Z|n1 = (Z|n)^<*Z.n1*> by FINSEQ_5:10;
    E1.n1 in rng E1 & rng E1 c= the carrier of FC
      by A7,A1,FUNCT_1:def 3;
    then reconsider E1n1=E1.n1 as Element of FC;
    f.n1 = f.n+E1n1 by A1,A3,A6,NAT_1:13;
    hence thesis by A8,A1,NAT_1:13,A6,RVSUM_2:31;
  end;
  P[n] from NAT_1: sch 2(A4,A5);
  then P[len Z] & Z|len Z=Z;
  hence thesis by A2,A1;
end;
