reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th10:
  (f,k) +...+ (f,n) <> 0 implies
     ex i st k <= i & i <=n & i in dom f
proof
  assume A1:(f,k) +...+ (f,n) <> 0;
  then A2:n >=k by Def1;
  then consider h such that
  A3:(f,k) +...+ (f,n) = Sum h and
  A4:len h = n-'k+1 and
  A5:h.(0+1) = f.(0+k) & ... & h.(n-'k+1) = f.(n-'k+k) by Th9;
  assume A6:for i st k <= i & i <=n holds not i in dom f;
  n-'k +1 >= 1 by NAT_1:11;
  then 1 in dom h by A4,FINSEQ_3:25;
  then A7: h.1 in rng h by FUNCT_1:def 3;
  rng h c= {0}
  proof
    let y be object;
    assume y in rng h;
    then consider x be object such that
    A8:x in dom h & h.x=y by FUNCT_1:def 3;
    reconsider x as Nat by A8;
    1<= x & x <= len h by A8,FINSEQ_3:25;
    then reconsider x1=x-1 as Nat;
    x1+1=x;
    then A9:x1<= n-'k by A8,FINSEQ_3:25,A4,XREAL_1:6;
    then A10:h.(x1+1)=f.(x1+k) by A5;
    n-'k = n- k by A2,XREAL_1:233;
    then k <= x1+k & x1+k <= n-'k+k & n-'k+k=n
      by A9,XREAL_1:6,NAT_1:11;
    then not x1+k in dom f by A6;
    then f.(x1+k)=0 by FUNCT_1:def 2;
    hence thesis by A10,A8,TARSKI:def 1;
  end;
  then h = (dom h)--> 0 by A7,ZFMISC_1:33,FUNCOP_1:9;
  then h = len h |->0 by FINSEQ_1:def 3;
  hence thesis by RVSUM_1:81,A3,A1;
end;
