holds:

1.  if $m=1$ the path $\gamma$ is reduced if it does not contain any repetition.

2.  for any simple cycle $\gamma'$ containing $(x_i,\ldots, x_{i+q})$ replacing all $(y,x,z)$ with $x\in (x_{i+1},\ldots, x_{i+q-1})\cap V_{j+m}$ by an $E_{j+m-1}$-path in $R(x)$ yields a reduced path $(x_i,\ldots, y,\ldots z,\ldots, x_{i+q})\subseteq V_j\cup\ldots \cup V_{j+m-1}$

Note that the definition and Lemma \[l:cycle\] immediately imply the following:

\[r:reducedpath\] Suppose that every reduced path from $a$ to $b$ contains $x$ and let $\gamma_1,\gamma_2$ be paths from $a$ to $x$ and from $x$ to $b$ respectively. Then the path $\gamma_1\gamma_2$ is reduced if and only if $\gamma_1$ and $\gamma_2$ are.

If $\gamma=(a,\ldots, b)$ is a reduced path and changes direction in $x\in V_k$, then either every reduced path from $a$ to $b$ contains $x$ or the path $(y,x,z)\subseteq \gamma$ can be replaced by an $E_{k-1}$-path (or by an $E_k$-path) $(y,\ldots z)$ such that $(a,\ldots, y,\ldots, z,\ldots b)$ is still reduced.

\[l:nicereplacement\] Suppose $\gamma=(a=x_0,\ldots, x_s=b)$ is a reduced path from $a$ to $b$ which can be completed into a simple cycle $(a,x_1,\ldots, b,\ldots,a)$. If $\gamma$ changes direction in $x\in V_k$, then there is a reduced path $\gamma'=(a=y_0,\ldots, y_t=b)$ not containing $x$ such that $x$ is on the $E_{k}$-(or $E_{k+1}$-) path 