why is that in proving the theorem
 

why is that in proving the theorem
xter,
it's just a sign convention. you can use any mathematically consistant system. many structural engineers prefer to call downwards deflection "+ve" and the corresponding bending moment positive also. mathematically however, a downwards load causing a downwards deflection is usually referred to as "-ve" in the conventional x-y coordinate system.
text books use either convention, depending usually on the author's preference.