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 Th15:
  k > len h implies (h,k) +...+ (h,n) = 0
proof
  assume A1:k >len h;
  per cases;
  suppose k>n;
    hence thesis by Def1;
  end;
  suppose k <=n;
    then consider w be complex-valued FinSequence such that
    A2:(h,k) +...+ (h,n) = Sum w & len w = n-'k+1 and
    A3:w.(0+1) = h.(0+k) & ... & w.(n-'k+1) = h.(n-'k+k) by Th9;
    set nk=n-'k+1,nk0 = nk |-> (0 qua Real);
    now let i such that A4: 1<= i & i <= nk;
      reconsider i1=i-1 as Nat by A4;
      i1+1=i;
      then i1 <= n-'k by A4,XREAL_1:6;
      then A5:w.(i1+1) = h.(i1+k) by A3;
      i1+k > 0+len h by A1,XREAL_1:8;
      then not i1+k in dom h by FINSEQ_3:25;
      hence w.i=nk0.i by FUNCT_1:def 2,A5;
    end;
    then w= nk|->0 by CARD_1:def 7,A2;
    then Sum w = nk*0 by RVSUM_1:80;
    hence thesis by A2;
  end;
end;
