# Welcome to MWiki

## Theorem of the month

### Counter-directional theorem

If the path $\displaystyle{ \gamma: [a, b[ \, \cap \, C \rightarrow V }$ with $\displaystyle{ C \subseteq \mathbb{R} }$ passes the edges of every $\displaystyle{ n }$-cube of side length $\displaystyle{ \iota }$ in the $\displaystyle{ n }$-volume $\displaystyle{ V \subseteq {}^{(\omega)}\mathbb{R}^{n} }$ with $\displaystyle{ n \in \mathbb{N}_{\ge 2} }$ exactly once, where the opposite edges in all two-dimensional faces of every $\displaystyle{ n }$-cube are traversed in reverse direction, but uniformly, then, for $\displaystyle{ D \subseteq \mathbb{R}^{2}, B \subseteq {V}^{2}, f = ({f}_{1}, ..., {f}_{n}): V \rightarrow {}^{(\omega)}\mathbb{R}^{n}, \gamma(t) = x, \gamma(\curvearrowright D t) = \curvearrowright B x }$ and $\displaystyle{ {V}_{\curvearrowright } := \{\curvearrowright B x \in V: x \in V, \curvearrowright B x \ne \curvearrowleft B x\} }$, it holds that

$\displaystyle{ \uparrow_{t \in [a,b[ \; \cap \; C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)\downarrow{Dt}}=\uparrow_{\begin{smallmatrix} (x,\curvearrowright B\,x) \\ \in V\times {{V}_{\curvearrowright}} \end{smallmatrix}}{f(x)\downarrow{Bx}}=\uparrow_{\begin{smallmatrix} t \in [a,b[ \; \cap \; C, \\ \gamma | {\partial{}^{\acute{n}}} V \end{smallmatrix}}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)\downarrow{Dt}}. }$

#### Proof:

If two arbitrary squares are considered with common edge of length d0 included in one plane, then only the edges of $\displaystyle{ V\times{V}_{\curvearrowright} }$ are not passed in both directions for the same function value. They all, and thus the path to be passed, are exactly contained in $\displaystyle{ {\partial}^{\acute{n}}V.\square }$