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 Th5:
  seq is bounded_above iff rng seq is bounded_above
proof
A1: seq is bounded_above implies rng seq is bounded_above
  proof
    assume seq is bounded_above;
    then consider t such that
A2: for n holds seq.n<t by SEQ_2:def 3;
    t is UpperBound of rng seq
    proof
      let r be ExtReal;
      assume r in rng seq;
      then ex n being object st n in dom seq & seq.n = r by FUNCT_1:def 3;
      hence r<=t by A2;
    end;
    hence thesis;
  end;
  rng seq is bounded_above implies seq is bounded_above
  proof
    assume rng seq is bounded_above;
    then consider t such that
A3:  t is UpperBound of rng seq;
A4: for r st r in rng seq holds r<=t by A3,XXREAL_2:def 1;
    now
      let n;
A5:   n in NAT by ORDINAL1:def 12;
      seq.n <= t by A4,FUNCT_2:4,A5;
      hence seq.n < t+1 by Lm1;
    end;
    hence thesis by SEQ_2:def 3;
  end;
  hence thesis by A1;
end;
