derivation of divergence from nabla operator

Question

For a two dimensional orthogonal curvilinear coordinate system $(t_1, t_2)$, we have the position vector $r$, where $h_i = | \frac{\partial r}{\partial t_i} |$ are the scale factors and $a_i$ are the unit vectors defined as $a_i =\frac{\partial r}{\partial t_i}/ | \frac{\partial r}{\partial t_i} |$. The $\nabla$ operator is commonly defined as $\nabla = \frac{a_1}{h_1}\frac{\partial}{\partial t_1} + \frac{a_2}{h_2}\frac{\partial}{\partial t_2}$. Then, for a vector field $F = f_1 a_1 + f_2 a_2$, the divergence can be seen as the dot product: $\nabla \cdot F$, which gives us $\nabla \cdot F = \frac{1}{h_1h_2}(\frac{\partial}{\partial t_1}(h_2 f_1)+\frac{\partial}{\partial t_2}(h_1 f_2))$. Could someome explain me what ae the steps to prove that $(\frac{a_1}{h_1}\frac{\partial}{\partial t_1} + \frac{a_2}{h_2}\frac{\partial}{\partial t_2})\cdot(f_1 a_1 + f_2 a_2)$ equals $\frac{1}{h_1h_2}(\frac{\partial}{\partial t_1}(h_2 f_1)+\frac{\partial}{\partial t_2}(h_1 f_2))$?