How to prove Slutsky matrix's symmetry for L=2

I don't understand how to prove slutsky matrix is symmetric for L=2 $$\frac{\partial x_1}{\partial p_2}+\frac{\partial x_1}{\partial w}\cdot x_2= \frac{\partial x_2}{\partial p_1}+\frac{\partial x_2}{\partial w}\cdot x_1$$

Let $$c(p, u)$$ be the expenditure function. The Hicksian demand for good $$j$$ is the derivative of $$c$$ with respect to $$p_j$$. $$\frac{\partial c(p,u)}{\partial p_j} = h_j(p,u).$$ From this, it follows (by Young's theorem) that: $$\frac{\partial h_j(p,u)}{\partial p_i} = \frac{\partial^2 c(p,u)}{\partial p_j \partial p_i} = \frac{\partial^2 c(p,u)}{\partial p_i \partial p_j} = \frac{\partial h_i(p,u)}{\partial p_j},$$ So the Hicksian cross price effects are symmetric. Using the Slutsky equation, we get: \begin{align*} &\frac{\partial x_i(p,m)}{\partial p_j} + \frac{\partial x_i(p,m)}{\partial m} x_i(p,m),\\ &= \frac{\partial h_i(p,u)}{\partial p_j},\\ &= \frac{\partial h_j(p,u)}{\partial p_i},\\ &= \frac{\partial x_j(p,m)}{\partial p_i} + \frac{\partial x_j(p,m)}{\partial m} x_j(p,m). \end{align*}