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 Th39:
  seq is bounded_above implies (superior_realsequence seq).0 = upper_bound seq
proof
  reconsider Y1 = {seq.k : 0 <= k} as Subset of REAL by Th29;
  (superior_realsequence seq).0 = upper_bound Y1 by Def5
    .= upper_bound seq by SETLIM_1:5;
  hence thesis;
end;
