Beta-Reduction exercise with pairs in Lambda Calculus

Question

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: $$ (<M,u>) <M,v> \simeq_{\beta} (((M)M)v)u$$

Answer 1

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.