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 Th35:
  seq is non-increasing implies for n for R being Subset of REAL
  st R = {seq.k : n <= k} holds upper_bound R = seq.n
proof
  assume
A1: seq is non-increasing;
  let n;
A2: now
    let r;
    assume r in {seq.k : n <= k};
    then consider r1 being Real such that
A3: r = r1 & r1 in {seq.k : n <= k};
    consider k1 such that
A4: r1 = seq.k1 and
A5: n <= k1 by A3;
    consider k being Nat such that
A6: n + k =k1 by A5,NAT_1:10;
    thus r <= seq.n by A1,A3,A4,A6,SEQM_3:7;
  end;
  let R be Subset of REAL;
A7: now
    let s;
A8: seq.n in {seq.k : n <= k};
    assume 0 < s;
    hence ex r st r in {seq.k : n <= k} & seq.n - s < r by A8,XREAL_1:44;
  end;
  assume
A9: R = {seq.k : n <= k};
  then
A10: R <> {} by SETLIM_1:1;
  R is bounded_above by A1,A9,Th31,LIMFUNC1:1;
  hence thesis by A9,A10,A2,A7,SEQ_4:def 1;
end;
