PROVE that the Newton's second law of motion is the real law of motion.

Dear student
Because from 2nd law you can relate 1st and 3rd law.
F = ma
from this you can say until and unless force is applied the body remains at rest (as acceleration will be zero).
F₁₂ = dp1/dt (rate of change of momentum)
F₂₁ = dp2/dt
adding them F₁₂ + F₂₁ = d/dt(p₁ + p₂)
when no external force acts total momentum will be constant. Then rate of change of momentum will be zero.
F₁₂ + F₂₁ = 0
F₁₂ = -F₂₁
To every action there is equal and opposite reaction.
Regards