Question 1: In $\mathbb{R}^2$, let $l_1$,$l_2$ be two parallel lines and $l_3$ another line which is not parallel to $l_1$. Given two measurable sets $E_1$ and $E_2$ in $l_1$ and $l_2$ respectively, both of which have positive (1-dimension) Lebesgue measure. Denote by $l_{u,v}$ the line passing through two points $u,v$. Now, we define the third set $E_3$ in $l_3$ as follows. $E_3=\lbrace{p\in l_3: p=l_{u,v}\cap l_3, u\in E_1, v\in E_2\rbrace}$.

Can we say that $E_3$ contains an interval (or say "a segment") in $l_3$?

EDIT. Joseph Van Name answered Question 1 affirmatively. The following Question 2 is a generalization of Question 1, which is not answered.

Question 2: In $\mathbb{R}^n$, let $L_1$,$L_2$ be two parallel (n-1)-dimensional hyperplanes and $L_3$ another (n-1)-hyperplane which is not parallel to $L_1$. Given two measurable sets $E_1$ and $E_2$ in $L_1$ and $L_2$ respectively, both of which have positive ((n-1)-dimension) Lebesgue measure. Denote by $l_{u,v}$ the line passing through two points $u,v$. Now, we define the third set $E_3$ in $L_3$ as follows. $E_3=\lbrace{p\in L_3: p=l_{u,v}\cap L_3, u\in E_1, v\in E_2\rbrace}$.

Can we say that $E_3$ contains an $(n-1)$-ball in $L_3$?

It seems that the result in "An elementary proof and an extension of a theorem of Steinhaus,Kuczma, Marcin E.; Kuczma, Marek, Glasnik Mat. Ser. III 6(26) (1971), 11–18." noted by Joseph Van Name does not apply to the general case $n\geq 3$ of Question 2.