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 Th10:
  for seq being sequence of REAL, Eseq being sequence of
  ExtREAL st seq = Eseq & seq is bounded_above
  holds upper_bound seq = sup rng Eseq
proof
  let seq be sequence of REAL, Eseq be sequence of ExtREAL such that
A1: seq = Eseq and
A2: seq is bounded_above;
  reconsider s = upper_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 x <= s
  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,RINFSUP1:7;
  end;
  then
A6: s is UpperBound of rng Eseq by XXREAL_2:def 1;
  then
A7: rng Eseq is bounded_above by XXREAL_2:def 10;
A8: s <= sup rng Eseq
  proof
    reconsider r1=sup rng Eseq as Element of REAL by A7,A4,XXREAL_2:57;
A9:  sup rng Eseq is UpperBound of rng Eseq by XXREAL_2:def 3;
    for n being Nat holds seq.n <= r1
     proof let n be Nat;
      n in NAT by ORDINAL1:def 12;
      hence thesis by A1,A3,FUNCT_1:3,XXREAL_2:def 1,A9;
     end;
    hence thesis by RINFSUP1:9;
  end;
  sup rng Eseq <= s by A6,XXREAL_2:def 3;
  hence thesis by A8,XXREAL_0:1;
end;
