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 Th54:
  seq is bounded implies upper_bound(inferior_realsequence seq) <= (
  superior_realsequence seq).n
proof
  reconsider Y2 = {seq.k2 : n <= k2} as Subset of REAL by Th29;
  assume
A1: seq is bounded;
A2: now
    let m;
    reconsider Y1 = {seq.k1 : m <= k1} as Subset of REAL by Th29;
    n <= n + m by NAT_1:11;
    then
A3: seq.(n+m) in Y2;
    Y2 is real-bounded by A1,Th33;
    then Y2 is bounded_above;
    then
A4: upper_bound Y2 >= seq.(n+m) by A3,SEQ_4:def 1;
    m <= n + m by NAT_1:11;
    then
A5: seq.(n+m) in Y1;
    Y1 is real-bounded by A1,Th33;
    then Y1 is bounded_below;
    then
A6: seq.(n+m) >= lower_bound Y1 by A5,SEQ_4:def 2;
    (inferior_realsequence seq).m = lower_bound Y1 by Def4;
    hence upper_bound Y2 >= (inferior_realsequence seq).m by A6,A4,XXREAL_0:2;
  end;
  (superior_realsequence seq).n = upper_bound Y2 by Def5;
  hence thesis by A2,Th9;
end;
