
theorem Th7:
for D be non empty set,
  Y be non empty with_non-empty_element FinSequenceSet of D,
 s be non-empty sequence of Y, n be Nat holds
  len(s.n) >= 1 & n < (Partial_Sums(Length s)).n
& (Partial_Sums(Length s)).n < (Partial_Sums(Length s)).(n+1)
proof
   let D be non empty set,
       Y be non empty with_non-empty_element FinSequenceSet of D,
       s be non-empty sequence of Y,
       n be Nat;
   defpred P[Nat] means $1 < (Partial_Sums(Length s)).$1;
A1:for k be Nat holds len(s.k) >= 1
   proof
    let k be Nat;
    dom s = NAT by FUNCT_2:def 1; then
    k in dom s by ORDINAL1:def 12;
    hence len(s.k) >= 1 by FINSEQ_1:20;
   end;
   (Partial_Sums(Length s)).0 = (Length s).0 by SERIES_1:def 1
    .= len(s.0) by Def3; then
A3:P[0];
A4:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A5: P[k];
A6: (Partial_Sums(Length s)).(k+1)
      = (Partial_Sums(Length s)).k + (Length s).(k+1) by SERIES_1:def 1;
    (Length s).(k+1) = len(s.(k+1)) by Def3;
    hence P[k+1] by A1,A6,A5,XREAL_1:8;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A3,A4);
   hence len(s.n) >= 1 & n < (Partial_Sums(Length s)).n by A1;
   (Partial_Sums(Length s)).(n+1)
    = (Partial_Sums(Length s)).n + (Length s).(n+1) by SERIES_1:def 1
   .= (Partial_Sums(Length s)).n + len(s.(n+1)) by Def3;
   hence thesis by XREAL_1:29;
end;
