reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;

theorem
  for X being non empty ext-real-membered set st ex y being UpperBound
  of X st y <> +infty holds X is bounded_above
proof
  let X be non empty ext-real-membered set;
  given y being UpperBound of X such that
A1: y <> +infty;
  per cases;
  suppose
A2: y = -infty;
    take 0;
    let x;
    assume
A3: x in X;
    X c= {-infty} by A2,Th51;
    hence thesis by A3,TARSKI:def 1;
  end;
  suppose
    y <> -infty;
    then y in REAL by A1,XXREAL_0:14;
    then reconsider y as Real;
    take y;
    thus thesis;
  end;
end;
