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 Th46:
  seq is bounded_below implies (inferior_realsequence seq).n = min
  ((inferior_realsequence seq).(n+1),seq.n)
proof
  reconsider Y2 = {seq.k : n+1 <= k} as Subset of REAL by Th29;
  reconsider Y1 = {seq.k : n <= k} as Subset of REAL by Th29;
  reconsider Y3 = {seq.n} as Subset of REAL;
A1: (inferior_realsequence seq).(n+1) = lower_bound Y2 by Def4;
  assume
A2: seq is bounded_below;
  then
A3: Y2 <> {} & Y2 is bounded_below by Th32,SETLIM_1:1;
A4: Y3 is bounded_below
  proof
    consider t such that
A5: for m holds t<seq.m by A2,SEQ_2:def 4;
    t is LowerBound of Y3
    proof
      let r be ExtReal;
      assume r in Y3;
      then r = seq.n by TARSKI:def 1;
      hence t <= r by A5;
    end;
    hence thesis;
  end;
  (inferior_realsequence seq).n = lower_bound Y1 by Def4;
  then (inferior_realsequence seq).n = lower_bound (Y2 \/ Y3) by SETLIM_1:2
    .= min(lower_bound Y2,lower_bound Y3) by A3,A4,SEQ_4:142
    .= min((inferior_realsequence seq).(n+1),seq.n) by A1,SEQ_4:9;
  hence thesis;
end;
