If a $=\left\langle a_{1}, a_{2}\right\rangle, \mathbf{b}=\left\langle\boldsymbol{b}_{1}, \boldsymbol{b}_{2}\right\rangle, \mathbf{c}=\left\langle\boldsymbol{c}_{1}, \boldsymbol{c}_{2}\right\rangle,$ and $\boldsymbol{m}$ and $n$ are real numbers, prove the stated property.
$$(m+n) \mathbf{a}=m \mathbf{a}+n \mathbf{a}$$