reserve n for Nat;

theorem
  for T being non empty 1-sorted, S being SetSequence of the carrier of
T, R being subsequence of S, n being Nat holds ex m being Element of
  NAT st m >= n & R.n = S.m
proof
  let T being non empty 1-sorted, S being SetSequence of the carrier of T, R
  being subsequence of S, n being Nat;
  consider NS being increasing sequence of NAT such that
A1: R = S * NS by VALUED_0:def 17;
   reconsider m = NS.n as Element of NAT;
  take m;
  thus m >= n by SEQM_3:14;
  n in NAT by ORDINAL1:def 12;
  then n in dom NS by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:13;
end;
