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

theorem Th32:
  x < y implies sup ].x,y.[ = y
proof
  assume
A1: x < y;
A2: for z being UpperBound of ].x,y.[ holds y <= z
  proof
    let z be UpperBound of ].x,y.[;
    for r st x < r & r < y holds r <= z
    by XXREAL_1:4,Def1;
    hence thesis by A1,XREAL_1:229;
  end;
  y is UpperBound of ].x,y.[ by Th24;
  hence thesis by A2,Def3;
end;
