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

theorem Th40:
  for X being ext-real-membered set holds X is non empty iff inf X <= sup X
proof
  let X be ext-real-membered set;
  thus X is non empty implies inf X <= sup X
  proof
    assume X is non empty;
    then consider x such that
A1: x in X by MEMBERED:8;
A2: x <= sup X by A1,Th4;
    inf X <= x by A1,Th3;
    hence thesis by A2,XXREAL_0:2;
  end;
  assume inf X <= sup X;
  hence thesis by Th38,Th39;
end;
