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 Th10:
  (for n holds r <= seq.n) iff seq is bounded_below & r <= lower_bound seq
proof
  thus (for n holds r <= seq.n) implies seq is bounded_below &
  r <= lower_bound seq
  proof
    assume
A1: for n holds r <= seq.n;
    now
      let m;
      r <= seq.m by A1;
      hence r -1 < seq.m by Lm1;
    end;
    hence
A2: seq is bounded_below by SEQ_2:def 4;
    now
      set r1=r - (lower_bound seq);
      assume r > lower_bound seq;
      then r1 > 0 by XREAL_1:50;
      then ex k st seq.k < (lower_bound seq) + r1 by A2,Th8;
      hence contradiction by A1;
    end;
    hence thesis;
  end;
  assume that
A3: seq is bounded_below and
A4: r <= lower_bound seq;
  now
    let n;
    lower_bound seq <= seq.n by A3,Th8;
    hence r <= seq.n by A4,XXREAL_0:2;
  end;
  hence thesis;
end;
