reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th31:
  seq is bounded_above implies for n for R being Subset of REAL st
  R = {seq.k : n <= k} holds R is bounded_above
proof
  assume
A1: seq is bounded_above;
  let n;
  set seq1 = seq ^\n;
  seq1 is bounded_above by A1,SEQM_3:27;
  then
A2: rng seq1 is bounded_above by Th5;
  let R be Subset of REAL;
  assume R = {seq.k : n <= k};
  hence thesis by A2,Th30;
end;
