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-%This is just a .tex file with a wishlist of functionalitys 
-
-
-Tupel struct
-
-\newtheorem{struct2}[theoremcount]{Struct2}
-
-\begin{theorem}
-    %Some Theorem.
-\end{theorem}
-\begin{proof}
-    %Wish for nice Function definition. --------------------- 
-
-    %Some Proof where we need a Function.
-    %Privisuly defined.
-    $n \in \naturals$.
-    There is a Set $A = \{A_{0}, ..., A_{n}\}$.
-    For all $i$ we have $A_{i} \subseteq X$.
-
-    Define function $f: X \to Y$,
-    \begin{align}
-        &x \mapsto \rfrac{y}{n} &; if \exists k \in \{1, ... n\}. x \in A_{k} \\
-        &x \mapsto 0 &; if x  \phi(x) \\ 
-        %phi is some fol formula
-
-        &x \mapsto \eta &; for \phi(x) and \psi(\eta) 
-
-        &x \mapsto \some_term(x)(u)(v)(w) &; \exist.u,v,w \psi(x)(u)(v)(w) \\ 
-        % here i see the real need of varibles that can be useds in the define term
-
-        &x \mapsto \some_else_term(x)   &; else 
-        % the else term would be great 
-
-        % the following axioms should be automaticly added.
-        % \dom{f} = X
-        % \ran{f} \subseteq Y
-        % f is function
-
-        % therefor we should add the prompt for a proof that f is well defined
-    \end{align}
-    \begin{proof_well_defined}
-        % we need to proof that f allways maps X to Y
-    \end{proof_well_defined}
-
-    % more proof but now i can use the function f
-
-    % --------------------------------------------------------
-    \begin{equation}
-        X=
-        \begin{cases}
-          0, & \text{if}\ a=1 \\
-          1, & \text{otherwise}
-        \end{cases}
-      \end{equation}
-
-
-
-
-\end{proof}
-
-
-%------------------------------------------
-% My wish for a new struct 
-% I think this could be just get implemented along with the old struct
-
-
-% If take we only take tupels,
-% then just a list of defining fol formulas should be enougth. 
-\begin{struct2}
-    We say $(X,O)$ is a topological space if
-    \begin{enumerate}
-        \item $X$ is a set. % or X = \{...\mid .. \} or X = \naturals ... or ...
-        \item $O \subseteq \pow X$.
-        \item $\forall x,y \in O. x \union y \in O$ 
-        \item %another formula
-        \item %....
-    \end{enumerate} 
-\end{struct2}
-
-
-% Then the proof of some thing is a structure is more easy.
-% Since if we have just a tupel and some formulas which has to be fufilled,
-% then we can make a proof as follows.
-
-\begin{struct2}
-    We say $(A,i,N)$ is a indexed set if
-    \begin{enumerate}
-        \item $f$ is a bijection from $N$ to $A$
-        \item $N \subseteq \naturals$
-    \end{enumerate}
-\end{struct2}
-
-
-\begin{theorem}
-    Let $A = \{ \{n\} \mid n \in \naturals \}$.
-    Let function $f: \naturals \to \pow{\naturals}$ such that,
-    \begin{algin}
-        \item x \mapsto \{x\} ; x \in \naturals
-    \end{algin}
-    Then $(A, f, \naturals)$ is a indexed set.
-\end{theorem}
-\begin{proof}
-    % Then we only need to proof that:
-    % \ran{f} = A
-    % \dom{f} = \naturals
-    % f is a bijection between $\naturals$ to $A$.
-\end{proof}
-