Immersed within the world of Physics (Quantum Theory in particular), I came across the following proof for connecting the starting LHS with the finishing RHS.
$ \begin{eqnarray} \int_{t_0}^tdt_1 \int_{t_0}^{t_1} dt_2V(t_1)V(t_2) &=& \int_{t_0}^tdt_2 \int_{t_2}^{t} dt_1V(t_1)V(t_2)\\ &=&\int_{t_0}^tdt_1 \int_{t_1}^{t} dt_2V(t_2)V(t_1)\\ &=&\frac{1}{2}\left[\int_{t_0}^tdt_1 \int_{t_1}^{t} dt_2 + \int_{t_0}^tdt_2 \int_{t_0}^{t_1} dt_1\right]V(t_1)V(t_2)\\ &=&\frac{1}{2}\int_{t_0}^tdt_1 \int_{t_0}^{t} dt_2V(t_1)V(t_2)\\ \end{eqnarray} $
Unfortunately, I've had limited mathematical training (for mathematical physics) and I don't understand the motivation for all of the steps involved. The second line is achieved from the first by change of variables $t_1 \longleftrightarrow t_2$, but intuitively, how would one get from the LHS 1st to RHS 1st and 2nd to 3rd to 4th?
Clearly, I'm missing the fundamentals, so all comments would be welcome.