I want to construct a CPA-secure schemes using two given schemes $\prod_1$ and $\prod_2$ if only one of them is CPA secure.
Taking suggestions from this answer, I am able construct a scheme as follows.
Given a message $m$, Generate a random string $r$, of same lenght as $m$.
Compute $r_2 = r \oplus m$
Encrypt $r$ using $\prod_1$ to get $c_1$ and $r_2$ using $\prod_2$ to get $c_2$.
Send both $c1$ and $c2$
The scheme must be CPA secure because both of the ciphertexts need to be decrypted to get back the message.
However, I am not able to prove that the resultant scheme is CPA secure other than intuition.
How do I prove it rigorously?