Beta-Reduction exercise with pairs in Lambda Calculus I'm doing some simple exercise about Lambda Calculus but i have doubt about this beta-reduction.
Let $$<u,v>= \lambda p((p)u)v$$ a pair in Lambda Calculus.
Prove that for every lambda term M you have that:
$$ (<M,u>) <M,v>  \simeq_{\beta} (((M)M)v)u$$ 

To prove that you can follow this steps:
$$(<M,u>) <M,v>  \simeq_{\alpha} (\lambda p((p)M)u) \space \lambda q((q)M)v \\ \simeq_{\beta} ((\lambda q((q)M)v)M)u\\ \simeq_{\beta} ((M)M)v)u$$
Q.E.D.
