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

theorem
  A is bounded_below iff inf A <> -infty
proof
  hereby
    assume
A1: A is bounded_below;
    per cases by A1,Th58;
    suppose
      A = {};
      hence inf A <> -infty by Th38;
    end;
    suppose
      A = {+infty};
      hence inf A <> -infty by Lm2;
    end;
    suppose
      inf A in REAL;
      hence inf A <> -infty;
    end;
  end;
  assume
A2: inf A <> -infty;
  per cases by A2,XXREAL_0:14;
  suppose
A3: inf A = +infty;
    take 0;
    +infty is LowerBound of A by A3,Def4;
    hence thesis by Th72;
  end;
  suppose
    inf A in REAL;
    then reconsider r = inf A as Real;
    take r;
    thus thesis by Def4;
  end;
end;
