reserve n,m,k for Element of NAT,
  x,X for set,
  A1 for SetSequence of X,
  Si for SigmaField of X,
  XSeq for SetSequence of Si;
reserve Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th11:
  for seq being sequence of REAL, Eseq being sequence of
  ExtREAL st seq = Eseq & seq is bounded_below
   holds lower_bound seq = inf rng Eseq
proof
  let seq be sequence of REAL, Eseq be sequence of ExtREAL such that
A1: seq = Eseq and
A2: seq is bounded_below;
  reconsider s=lower_bound seq as R_eal by XXREAL_0:def 1;
A3: dom Eseq = NAT by FUNCT_2:def 1;
A4: rng Eseq <> {+infty}
  proof
    assume rng Eseq = {+infty};
    then reconsider k1 = +infty as Element of rng Eseq by TARSKI:def 1;
    consider n1 being object such that
A5: n1 in NAT and
    Eseq.n1 = k1 by A3,FUNCT_1:def 3;
    reconsider n1 as Element of NAT by A5;
    seq.n1 = k1 by A1;
    hence contradiction;
  end;
  for x being ExtReal holds x in rng Eseq implies s <= x
  proof
    let x be ExtReal;
    assume x in rng Eseq;
    then ex n1 being object st n1 in NAT & Eseq.n1 = x by A3,FUNCT_1:def 3;
    hence thesis by A1,A2,RINFSUP1:8;
  end;
  then
A6: s is LowerBound of rng Eseq by XXREAL_2:def 2;
  then
A7: rng Eseq is bounded_below by XXREAL_2:def 9;
A8: inf rng Eseq <= s
  proof
    reconsider r1=inf rng Eseq as Element of REAL by A7,A4,XXREAL_2:58;
A9:    inf rng Eseq is LowerBound of rng Eseq by XXREAL_2:def 4;
   for n being Nat holds r1 <= seq.n
    proof let n be Nat;
      n in NAT by ORDINAL1:def 12;
     hence thesis by A1,A3,FUNCT_1:3,XXREAL_2:def 2,A9;
    end;
    hence thesis by RINFSUP1:10;
  end;
  s <= inf rng Eseq by A6,XXREAL_2:def 4;
  hence thesis by A8,XXREAL_0:1;
end;
