reserve n for Nat,
  k for Integer;

theorem Th13:
  for a,b,s being complex-valued FinSequence st len s > 0 &
  len a = len s & len b = len s &
  (for n being Nat st 1 <= n & n <= len s holds s.n = a.n + b.n) &
  (for k being Nat st 1 <= k & k < len s holds b.k = -(a.(k+1)))
  holds Sum s = (a.1) + (b.(len s))
proof
  defpred P[FinSequence of COMPLEX] means len $1 > 0 implies for a,b being
  FinSequence of COMPLEX st len a = len $1 & len b = len $1 &
  (for n being Nat st 1 <= n & n <= len $1 holds $1.n = a.n + b.n) &
  (for k being Nat st 1 <= k & k < len $1 holds b.k = -(a.(k+1))) holds
  Sum $1 = a.1 + b.(len $1);
A1: for p being FinSequence of COMPLEX, x being Element of COMPLEX st P[p]
    holds P[p^<*x*>]
  proof
    let p be FinSequence of COMPLEX, x be Element of COMPLEX such that
A2: P[p];
    set t = p ^ <*x*>;
    assume len t > 0;
    let a,b be FinSequence of COMPLEX such that
A3: len a = len t and
A4: len b = len t and
A5: for n being Nat st 1 <= n & n <= len t holds t.n = a.n + b.n and
A6: for k being Nat st 1 <= k & k < len t holds b.k = -(a.(k+1));
A7: Sum t = (Sum p) + x by RVSUM_2:31;
    per cases;
    suppose
A8:   len p = 0;
      reconsider egy = 1 as Nat;
      p = {} by A8;
      then
A9:   t = <*x*> by FINSEQ_1:34;
      then egy <= len t by FINSEQ_1:39;
      then
A10:  t.egy = a.egy + b.egy by A5;
A11:  p = {} by A8;
      len t = 1 by A9,FINSEQ_1:39;
      hence thesis by A7,A11,A9,A10,GR_CY_1:3;
    end;
    suppose
A12:  len p > 0;
      set m = len p;
      set a9 = a|m;
A13:  a9.1 = a.1
      proof
        reconsider egy = 1 as Element of NAT;
        0 + 1 = 1;
        then egy <= len p by A12,INT_1:7;
        hence thesis by FINSEQ_3:112;
      end;
      set b9 = b|m;
A14:  b.(len p) = b9.(len p) by FINSEQ_3:112;
A15:  for n being Nat st 1 <= n & n < len p holds b9.n = -(a9.(n+1))
      proof
        let n be Nat such that
A16:    1 <= n and
A17:    n < len p;
        reconsider n as Element of NAT by ORDINAL1:def 12;
A18:    b9.n = b.n by A17,FINSEQ_3:112;
        n + 1 <= len p by A17,INT_1:7;
        then
A19:    a9.(n+1) = a.(n+1) by FINSEQ_3:112;
        len p <= len t by CALCUL_1:6;
        then n < len t by A17,XXREAL_0:2;
        hence thesis by A6,A16,A18,A19;
      end;
A20:  for n being Nat st 1 <= n & n <= len p holds p.n = a9.n + b9.n
      proof
        let n be Nat such that
A21:    1 <= n and
A22:    n <= len p;
        dom p = Seg len p by FINSEQ_1:def 3;
        then
A23:    n in dom p by A21,A22;
        len p <= len t by CALCUL_1:6;
        then
A24:    n <= len t by A22,XXREAL_0:2;
        reconsider n as Element of NAT by ORDINAL1:def 12;
        p.n = t.n by A23,FINSEQ_1:def 7
          .= a.n + b.n by A5,A21,A24
          .= a9.n + b.n by A22,FINSEQ_3:112
          .= a9.n + b9.n by A22,FINSEQ_3:112;
        hence thesis;
      end;
A25:  1 <= len p by A12,Lm1;
      len <*x*> = 1 by FINSEQ_1:39;
      then
A26:  len t = (len p) + 1 by FINSEQ_1:22;
      0 + len p = len p;
      then len p < len t by A26,XREAL_1:6;
      then
A27:  b.(len p) = -(a.(len p + 1)) by A6,A25;
      0 + 1 = 1;
      then
A28:  1 <= len t by A26,XREAL_1:6;
A29:  x = t.(len p + 1) by FINSEQ_1:42
        .= -(b9.(len p)) + b.(len t) by A5,A26,A28,A27,A14;
      m <= len b by A4,CALCUL_1:6;
      then
A30:  len b9 = len p by FINSEQ_1:59;
      m <= len a by A3,CALCUL_1:6;
      then len a9 = len p by FINSEQ_1:59;
      then Sum p = a9.1 + b9.(len p) by A2,A12,A30,A20,A15;
      hence thesis by A7,A13,A29;
    end;
  end;
A31: P[<*>COMPLEX];
A32: for p being FinSequence of COMPLEX holds P[p] from FINSEQ_2:sch 2(A31,A1);
  let a,b,s be complex-valued FinSequence;
A33: a is FinSequence of COMPLEX & b is FinSequence of COMPLEX &
     s is FinSequence of COMPLEX by FINSEQ_1:107;
  assume len s > 0 & len a = len s & len b = len s &
  (for n being Nat st 1 <= n & n <= len s holds s.n = a.n + b.n) &
  for k being Nat st 1 <= k & k < len s holds b.k = -(a.(k+1));
  hence thesis by A32,A33;
end;
