\ (L \ N))]\tr_{\mu}P\tr_{\rho}M'[(\a \ L]:=(\a \
    (L \ N))]$ for some $P$.

3.  Let $M\tr_{\rho}P\tr_{\mu}N$. Then $ M\tr_{\mu}Q\tr^*_{\rho}N$ for some $Q$.

By induction on $M$.

\[ch1:ppr\] Let $M\tr^*_{\rho}P\tr_{\b \m}N$. Then $M\tr_{\b \m}Q\tr^*_{\rho}N$ for some $Q$.

By induction on $lg(M\tr_{\rho}^*P)$.

Commutation lemmas
------------------

The goal of this section is lemma \[diag\*\] below. Its proof necessitates some preliminary lemmas.

\[diag\]

1.  If $M \tr_{\r} P$ and $M \tr_{\r\th} Q$, then $P = Q$ or $P
    \tr_{\r\th} N$ and $Q \tr_{\r} N$ for some $N$.

2.  If $M \tr_{\r} P$ and $M \tr_{\b\m} Q$, then $P \tr_{\b\m} N$ and $Q \tr_{\r}^* N$ for some $N$.

By simple case analysis.

\[new\]

1.  If $M \tr_{\r}^* P$ and $M \tr_{{\r\th}^1} Q$, then $P
    \tr_{{\r\th}^1} N$ and $Q \tr_{\r}^* N$ for some $N$.

2.  If $M \tr_{\r}^* P$ and $M \tr_{\r\th}^* Q$, then $P
    \tr_{\r\th}^* N$ and $Q \tr_{\r}^* N$ for some $N$.

3.  If $M \tr_{\r}^* P$ and $M \tr_{\b\m} Q$, then $P \tr_{\b\m} N$ and $Q \tr_{\r}^* N$ for some $N$.

<!-- -->

1.  By induction on $\eta_{\r}(M)$. Use (1) of lemma \[diag\].

2.  By induction on $lg(M \tr_{\r\th}^* Q)$. Use (1).

3.  By induction on $\eta_{\r}(M)$. Use (2) of lemma \[diag\].

\[diag\*\] 