let x, y be ext-real number ; :: thesis: ( x < y implies sup ].x,y.] = y )
A: y is UpperBound of ].x,y.] by Th22;
assume Z: x < y ; :: thesis: sup ].x,y.] = y
for z being UpperBound of ].x,y.] holds y <= z

proof

let z be UpperBound of ].x,y.]; :: thesis: y <= z
y in ].x,y.] by Z, XXREAL_1:2;
hence y <= z by Def1; :: thesis: verum

end;

hence sup ].x,y.] = y by A, Def3; :: thesis: verum