Cumulative distribution function - sum of two independen random variables Given the following cdfs of two independent random variable:

Find cdf $F_w$ which is the sum of those two independent variables cdfs. How to find this cdf? I do not know how to deal with those type of questions, because i do not think it is going to be a simple sum of those. 
QUESTION STILL NOT FULLY ANSWERED
my attempt: $$\int_{-\infty}^{\infty} 0.2(z-y)0.1y dy=\int_{-\infty}^{\infty}0.02(zy-y^2)dy=\left[0.01zy^2-\frac{1}{3}y^3 \right]_0^1=0.01z-1/3$$ and this will be for $x\in [0,1)$ Whereas for $x \ge 1$ we have 1 and for $x<0$ we have $0$

You have $F_1(x) = (0.6+0.2x) \mathbf 1_{x\in[0;1)}+\mathbf 1_{x\in[1;\infty)} \\ F_2(y) = (0.7+0.1y)\mathbf 1_{y\in[0;1)}+\mathbf 1_{y\in[1;\infty)}$
Both CDF have step discontinuities at arguments of $0$ and $1$, indicating massive points.
$$\mathsf P(X_1=0) = 0.6,~ \mathsf P(X_1=1)=0.2,~ \mathsf P(0<X_1\leqslant x <1)=0.2x
\\
\mathsf P(X_2=0) = 0.7,~ \mathsf P(X_2=1)=0.2,~ \mathsf P(0<X_2 \leqslant y<1)=0.1y$$
Then $\mathsf P(X_1+X_2\leq z) = \begin{cases} 0 & : z\in (-\infty;0)
\\ 0.6\cdot 0.7 + 0.6\cdot 0.1z +0.2z\cdot 0.7+0.2\cdot 0.1 z^2/2 & : z\in[0;1) \\ 0.6+0.4\cdot 0.7 & : z = 1 \\ \text{can you complete?} & : z\in (1; 2) \\ 1 & z\in [2;\infty)\end{cases} $ 
