reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;

theorem
  for F,G being FinSequence of the carrier of V st len F = len G & (for
k being Nat,v being VECTOR of V st k in dom F & v = G.k holds F.k =
  z * v) holds Sum(F) = z * Sum(G)
proof
  let F,G be FinSequence of the carrier of V;
  defpred P[set] means for H,I being FinSequence of the carrier of V st len H
= len I & len H = $1 & (for k being Nat, v being VECTOR of V st k in
  Seg len H & v = I.k holds H.k = z * v) holds Sum(H) = z * Sum(I);
A1: dom F = Seg len F by FINSEQ_1:def 3;
  now
    let n be Nat;
    assume
A2: for H,I being FinSequence of the carrier of V st len H = len I &
len H = n & for k being Nat, v being VECTOR of V st k in Seg len H &
    v = I.k holds H.k = z * v holds Sum(H) = z * Sum(I);
    let H,I be FinSequence of the carrier of V;
    assume that
A3: len H = len I and
A4: len H = n + 1 and
A5: for k being Nat, v being VECTOR of V st k in Seg len H
    & v = I.k holds H.k = z * v;
    reconsider p = H | (Seg n),q = I | (Seg n) as FinSequence of the carrier
    of V by FINSEQ_1:18;
A6: n <= n + 1 by NAT_1:12;
    then
A7: len q = n by A3,A4,FINSEQ_1:17;
A8: len p = n by A4,A6,FINSEQ_1:17;
A9: now
      len p <= len H by A4,A6,FINSEQ_1:17;
      then
A10:  Seg len p c= Seg len H by FINSEQ_1:5;
A11:  dom p = Seg n by A4,A6,FINSEQ_1:17;
      let k be Nat;
      let v be VECTOR of V;
      assume that
A12:  k in Seg len p and
A13:  v = q.k;
      dom q = Seg n by A3,A4,A6,FINSEQ_1:17;
      then I.k = q.k by A8,A12,FUNCT_1:47;
      then H.k = z * v by A5,A12,A13,A10;
      hence p.k = z * v by A8,A12,A11,FUNCT_1:47;
    end;
    n + 1 in Seg(n + 1) by FINSEQ_1:4;
    then n +1 in dom H & n +1 in dom I by A3,A4,FINSEQ_1:def 3;
    then reconsider v1 = H.(n + 1),v2 = I.(n + 1) as VECTOR of V by FUNCT_1:102
;
A14: v1 = z * v2 by A4,A5,FINSEQ_1:4;
A15: dom q = Seg len q by FINSEQ_1:def 3;
    dom p = Seg len p by FINSEQ_1:def 3;
    hence Sum(H) = Sum(p) + v1 by A4,A8,RLVECT_1:38
      .= z * Sum(q) + z * v2 by A2,A8,A7,A9,A14
      .= z * (Sum(q) + v2) by Def2
      .= z * Sum(I) by A3,A4,A7,A15,RLVECT_1:38;
  end;
  then
A16: for n being Nat st P[n] holds P[n+1];
  now
    let H,I be FinSequence of the carrier of V;
    assume that
A17: len H = len I and
A18: len H = 0 and
    for k being Nat,v being VECTOR of V st k in Seg len H & v =
    I.k holds H.k = z * v;
    Sum(H) = 0.V by A18,Lm2;
    hence Sum(H) = z * Sum(I) by A17,A18,Lm2,Th1;
  end;
  then
A19: P[0];
  for n being Nat holds P[n] from NAT_1:sch 2(A19,A16);
  hence thesis by A1;
end;
