Proof

By induction on the definition of $\stackrel{{R}}{\leftrightarrow}$.

1. $M \stackrel{{R}}{\leftrightarrow} N$ because $M~R~N$. Then chooze $Z \equiv N$.

2. $M \stackrel{{R}}{\leftrightarrow} N$ because $N \stackrel{{R}}{\leftrightarrow} M$. Trivial.

3. $M \stackrel{{R}}{\leftrightarrow} N$ because $M \stackrel{{R}}{\leftrightarrow} P$ and $P \stackrel{{R}}{\leftrightarrow} N$. By the induction hypothesis, we have $Y$ such that $M~R~Y$ and $P~R~Y$ and $Z$ such that $P~R~Z$ and $N~R~Z$.

   Since $P~R~Y$, $P~R~Z$ and $R$ is Church-Rosser, there is a term $W$ such that $Y~R~W$ and $Z~R~W$.

   But then $M~R~W$ and $N~R~W$.

$\Box$

We have the following useful Corollary: