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

theorem
  sup A = inf A implies A = {inf A}
proof
  assume
A1: sup A = inf A;
  then A c= [.inf A,inf A.] by Th69;
  then
A2: A c= {inf A} by XXREAL_1:17;
  A <> {} by A1,Th38,Th39;
  hence thesis by A2,ZFMISC_1:33;
end;
