By Newton's Third Law of Motion,

$\mathbf{F}_{12}=-\mathbf{F}_{21}$

By Newton's Second Law of Motion,

$\begin{aligned} \mathbf{F}_{12}&=-\mathbf{F}_{21}\\ \dot{\mathbf{p}}_{1}&=-\dot{\mathbf{p}}_{2}\\ \int_{t_1}^{t_2}\dot{\mathbf{p}}_{1}\,dt&=-\int_{t_1}^{t_2}\dot{\mathbf{p}}_{2}\,dt\\ \Delta\mathbf{p}_{1}&=-\Delta\mathbf{p}_{2}\\ \Delta(m\mathbf{v}_{1})&=-\Delta(m_2\mathbf{v}_{2})\\ m_1\Delta(\mathbf{v}_{1})&=-m_2\Delta(\mathbf{v}_{2})\\ m_1(\mathbf{v}_{1}-\mathbf{u}_{1})&=-m_2(\mathbf{v}_{2}-\mathbf{u}_{2})\\ m_1\mathbf{v}_{1}-m_1\mathbf{u}_{1}&=m_2\mathbf{u}_{21}-m_2\mathbf{v}_{2}\\ m_1\mathbf{v}_{1}+m_2\mathbf{v}_{2}&=m_2\mathbf{u}_{2}+m_1\mathbf{u}_{1}\\ \sum\mathbf{p}_{f}&=\sum\mathbf{p}_{i}\quad\blacksquare \end{aligned}$