$M,N$ be such that $M\tr^*_{\b \m
\rho \theta}N$ and $lg_{\b\m}(M\tr^*_{\b \m \rho \theta} N) \geq
1$. Then $M\tr_{\b \m }^+P\tr^*_{\rho \theta}N$ for some $P$.

This is proved in two steps. First we show that the $\tr_{\th}$-reduction can be postponed w.r.t. to $\tr_{\b\m\rho}$ (theorem \[ch1:ppt\]). Then we show that the $\tr_{\rho}$-rule can be postponed w.r.t. the remaining rules (theorem \[ch1:ppr\]).

Say that $P \tr_{\mu_0}P' $ if $P= (\mu \a M\;N),
  P'=\mu \a M[(\a \ L]:=(\a \ (L \ N))]$ and $\a$ occurs at most once in $M$

\[ch1:bth\]

1.  Assume $M\tr_{\th}P\tr_{\b\m}N$. Then either $M\tr_{\b\m}Q\tr^*_{\th}N$ for some $Q$ or $M\tr_{\mu_0}R\tr_{\b\m}Q\tr_{\th}N$ for some $R$, $Q$.

2.  Let $M\tr_{\th}P\tr_{\mu_0}N$. Then either $M\tr_{\mu_0}Q\tr_{\th}N$ for some $Q$ or $M\tr_{\mu_0}R\tr_{\mu_0}Q\tr_{\th}N$ for some $R$, $Q$.

3.  Let $M\tr_{\th}P\tr_{\rho}N$. Then $M\tr_{\rho}Q\tr_{\th}N$.

By induction on $M$.

\[ch1:pth\]

Let $M\tr^*_{\th}P\tr_{\mu_0}N$. Then, $
M\tr^*_{\mu_0}Q\tr^*_{\th}N$ for some $Q$ such that $lg(M\tr_{\th}^*P)=lg(Q\tr_{\th}^*N)$.

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

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

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

\[ch1:brh\]

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

2.  Let $M$, $M'$, $N$ be such that $M\tr_{\rho}M'$ and $\a \notin Fv(N)$. Then either $M[(\a \
    L]:=(\a \ (L \ N))]\tr_{\rho}M'[(\a \ L]:=(\a \ (L \ N))]$ or $M[(\a \ L]:=(\a 