diff --git a/trees/hopf-0002.tree b/trees/hopf-0002.tree
index 8768e4e..870b0be 100644
--- a/trees/hopf-0002.tree
+++ b/trees/hopf-0002.tree
@@ -5,16 +5,9 @@
 \p{
 Let #{V} be a linear space of finite dimension #{n}. Let lower case #{x_i} denote elements of #{V}, which we will call also letters. We define a bracket as an alternating multilinear scalar valued function
 ##{
-\begin{gathered}
-{[, \ldots, .]: V \times \ldots \times V \rightarrow \mathbb{k}} \\
-{\left[x_1, \ldots, x_n\right]=\operatorname{sign}(p)\left[x_{p(1)}, \ldots, x_{p(n)}\right]} \\
-{\left[x_1, \ldots, \alpha x_r+\beta y_r, \ldots, x_n\right]=\alpha\left[x_1, \ldots, x_r, \ldots, x_n\right]+\beta\left[x_1, \ldots, y_r, \ldots, x_n\right]}
-\end{gathered}
-}
-#{n}-factors
-##{
 \begin{aligned}
-{\left[x_1, \ldots, x_n\right] } & =\operatorname{sign}(p)\left[x_{p(1)}, \ldots, x_{p(n)}\right] \\
+[, \ldots, .] & : V \times \ldots \times V \rightarrow \mathbb{k} \quad (n\text{-factors})   \\
+{\left[x_1, \ldots, x_n\right]} & =\operatorname{sign}(p)\left[x_{p(1)}, \ldots, x_{p(n)}\right]  \\
 {\left[x_1, \ldots, \alpha x_r+\beta y_r, \ldots, x_n\right] } & =\alpha\left[x_1, \ldots, x_r, \ldots, x_n\right]+\beta\left[x_1, \ldots, y_r, \ldots, x_n\right]
 \end{aligned}
 }}