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

theorem Th30:
  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.];
    y in ].x,y.] by A1,XXREAL_1:2;
    hence thesis by Def1;
  end;
  y is UpperBound of ].x,y.] by Th22;
  hence thesis by A2,Def3;
end;
