
theorem Th11:
for D be non empty set, Y be with_non-empty_element FinSequenceSet of D,
    s be non-empty sequence of Y
 ex N be increasing sequence of NAT st
  for k be Nat holds N.k = (Partial_Sums(Length s)).k - 1
proof
   let D be non empty set, Y be with_non-empty_element FinSequenceSet of D,
       s be non-empty sequence of Y;
   defpred P[Nat,Nat] means $2 = (Partial_Sums(Length s)).$1 - 1;
A1:for k be Element of NAT ex n be Element of NAT st P[k,n]
   proof
    let k be Element of NAT;
    reconsider n = (Partial_Sums(Length s)).k - 1 as Element of NAT
       by Th7,NAT_1:20;
    take n;
    thus thesis;
   end;
   consider N be Function of NAT,NAT such that
A2: for k be Element of NAT holds P[k,N.k] from FUNCT_2:sch 3(A1);
A3:for k be Nat holds N.k = (Partial_Sums(Length s)).k - 1
   proof
    let k be Nat;
    k is Element of NAT by ORDINAL1:def 12;
    hence thesis by A2;
   end;
   for n be Nat holds N.n < N.(n+1)
   proof
    let n be Nat;
    (Partial_Sums(Length s)).n - 1 < (Partial_Sums(Length s)).(n+1) - 1
      by Th7,XREAL_1:9; then
    N.n < (Partial_Sums(Length s)).(n+1) - 1 by A3;
    hence N.n < N.(n+1) by A3;
   end; then
   reconsider N as increasing sequence of NAT by VALUED_1:def 13;
   take N;
   thus thesis by A3;
end;
