
theorem Th6:
  for F,G being FinSequence of F_Complex st len G = len F & for i
being Element of NAT st i in dom G holds G/.i = (F/.i)*' holds Sum G = (Sum F)
  *'
proof
  defpred P[Nat] means for F,G being FinSequence of F_Complex st len G = len F
  & len F = $1 & for i being Element of NAT st i in dom G holds G/.i = (F/.i)*'
  holds Sum G = (Sum F)*';
  let F,G be FinSequence of F_Complex;
  assume that
A1: len G = len F and
A2: for i being Element of NAT st i in dom G holds G/.i = (F/.i)*';
A3: now
    let k be Nat;
    assume
A4: P[k];
    now
      let F,G be FinSequence of F_Complex;
      assume that
A5:   len F = len G and
A6:   len F = k+1 and
A7:   for i being Element of NAT st i in dom G holds G/.i=(F/.i)*';
      set G1 = G|(Seg k);
      reconsider G1 as FinSequence by FINSEQ_1:15;
      reconsider G1 as FinSequence of F_Complex by A5,A6,Lm1;
A8:   G = G1^<*G/.(k+1)*> by A5,A6,Lm1;
      set F1 = F|(Seg k);
      reconsider F1 as FinSequence by FINSEQ_1:15;
      reconsider F1 as FinSequence of F_Complex by A6,Lm1;
A9:   len F1 = k by A6,Lm1;
A10:  len G1 = k by A5,A6,Lm1;
      then
A11:  dom G1 = Seg len F1 by A9,FINSEQ_1:def 3
        .= dom F1 by FINSEQ_1:def 3;
      1 <= k + 1 by NAT_1:11;
      then
A12:  k + 1 in dom G by A5,A6,FINSEQ_3:25;
A13:  F = F1^<*F/.(k+1)*> by A6,Lm1;
A14:  dom G = Seg len F by A5,FINSEQ_1:def 3
        .= dom F by FINSEQ_1:def 3;
A15:  now
        let i be Element of NAT;
        assume
A16:    i in dom G1;
A17:    dom G1 c= dom G by A5,A6,Lm1;
        then
A18:    F/.i = F.i by A14,A16,PARTFUN1:def 6
          .= F1.i by A13,A11,A16,FINSEQ_1:def 7
          .= F1/.i by A11,A16,PARTFUN1:def 6;
        thus G1/.i = G1.i by A16,PARTFUN1:def 6
          .= G.i by A8,A16,FINSEQ_1:def 7
          .= G/.i by A16,A17,PARTFUN1:def 6
          .= (F1/.i)*' by A7,A16,A17,A18;
      end;
      thus (Sum F)*' = (Sum F1 + Sum<*F/.(k+1)*>)*' by A13,RLVECT_1:41
        .= (Sum F1)*' + (Sum<*F/.(k+1)*>)*' by COMPLFLD:51
        .= Sum G1 + (Sum<*F/.(k+1)*>)*' by A4,A10,A9,A15
        .= Sum G1 + (F/.(k+1))*' by RLVECT_1:44
        .= Sum G1 + G/.(k+1) by A7,A12
        .= Sum G1 + Sum<*G/.(k+1)*> by RLVECT_1:44
        .= Sum G by A8,RLVECT_1:41;
    end;
    hence P[k+1];
  end;
  now
    let F,G be FinSequence of F_Complex;
    assume that
A19: len F = len G and
A20: len F = 0 and
    for i being Element of NAT st i in dom G holds G/.i=(F/.i)*';
    F = <*>(the carrier of F_Complex) by A20;
    then Sum(F) = 0.F_Complex by RLVECT_1:43;
    then
A21: Sum(F) = (0.F_Complex)*' by Lm2,COMPLEX1:38;
    G = <*>(the carrier of F_Complex) by A19,A20;
    hence Sum G = Sum(F)*' by A21,RLVECT_1:43;
  end;
  then
A22: P[0];
  for k be Nat holds P[k] from NAT_1:sch 2(A22,A3);
  hence thesis by A1,A2;
end;
