Proving That $R$ is an Equivalence Relation: Is This Solution Proof for Transitivity Incorrect?

Question

I have the following problem:

The solution proof for transitivity is as follows:

I suspect that this solution is incorrect. Specifically, I think that the very last part is incorrect.

My Solution

Let $((a, b), (c, d)), ((c, d), (e, f)) \in R$.

$\therefore (a, c), (c, e) \in R_1$ and $(b, d), (d, f) \in R_2$. (By the hypothesis.)

$\therefore (a, e) \in R_1$ and $(b, f) \in R_2$. (Since $R_1, R_2$ are equivalence relations.)

$\therefore ((a, b), (e, f)) \in R$ (By the hypothesis.)

$Q.E.D.$

I would greatly appreciate it if people could please take the time to review this and provide feedback. If I have made any errors, then I would appreciate clarification.

Answer 1

Your proof of transitivity is flawless. You have made it very clear at every point of the proof.