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 Th6:
  seq is bounded_below iff rng seq is bounded_below
proof
A1: seq is bounded_below implies rng seq is bounded_below
  proof
    assume seq is bounded_below;
    then consider t such that
A2: for n holds t<seq.n by SEQ_2:def 4;
    t is LowerBound 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 t<=r by A2;
    end;
    hence thesis;
  end;
  rng seq is bounded_below implies seq is bounded_below
  proof
    assume rng seq is bounded_below;
    then consider t such that
A3:  t is LowerBound of rng seq;
A4: for r st r in rng seq holds t<=r by A3,XXREAL_2:def 2;
    now
      let n;
A5:   n in NAT by ORDINAL1:def 12;
      t <= seq.n by A4,FUNCT_2:4,A5;
      hence t - 1 < seq.n by Lm1;
    end;
    hence thesis by SEQ_2:def 4;
  end;
  hence thesis by A1;
end;
