# Sum of absolutely continuous independent random variables

I'm facing the proof of a theorem stating that the sum of two absolutely continuous independent random variables is a new absolutely continuous random variable whose density is given by the convolution of the density functions of the variables. At the beginning I find this statement that I'm struggling to understand (consider that before this theorem we onlybhad the definition of A.C. random variable and independence of random variables).

Let $$X_1, X_2$$ be two absolutely continuous random variables, and $$\rho_1(t), \rho_2(t)$$ be their density. Let $$Y=X_1+X_2$$. Then $$\mathbb{P}( Y \leq t )= \mathbb{P}( X_1+ X_2 \leq t ) = \int\limits_{-\infty}^{+\infty} \rho_1(t_1) \cdot \int\limits_{-\infty}^{t-t_1} \rho_2(t_2) dt_2 dt_1$$

How did we get this last step?

$$P\{X_1+X_2 \leq t|X_1=t_1)=P\{X_2 \leq t -X_1|X_1=t_1)$$ $$=\int_{-\infty} ^{t-t_1} \rho_2(t_2)dt_2.$$ Now integrate w.r.t the distribution of $$X_1$$.