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## Theorem of the month

### Universal multistep theorem

For $\displaystyle{ n \in {}^{\nu}\mathbb{N}_{\le p}, k, m, p \in {}^{\nu}\mathbb{N}^{*}, d_{\curvearrowright B} x \in\, ]0, 1[, x \in [a, b] \subseteq {}^{\omega}\mathbb{R}, y : [a, b] \rightarrow {}^{\omega}\mathbb{R}^q, f : [a, b] \times {}^{\omega}\mathbb{R}^{q \times n} \rightarrow {}^{\omega}\mathbb{R}^q, g_k(\curvearrowright B x) := g_{\acute{k}}(x) }$ and $\displaystyle{ g_0(a) = f((\curvearrowleft B)a, y_0, ... , y_{\acute{n}}) }$, the Taylor series of the initial value problem $\displaystyle{ y^\prime(x) = f(x, y((\curvearrowright B)^0 x), ... , y((\curvearrowright B)^{\acute{n}} x)) }$ of order $\displaystyle{ n }$ implies

$\displaystyle{ y(\curvearrowright B x) = y(x) - d_{\curvearrowright B}x\sum\limits_{k=1}^{p}{i^{2k} g_{p-k}((\curvearrowright B) x)\sum\limits_{m=k}^{p}{\widehat{m!}\binom{\acute{m}}{\acute{k}}}} + \mathcal{O}((d_{\curvearrowright B} x)^{\grave{p}}).\square }$

### Goldbach’s theorem

Every even whole number greater than 2 is the sum of two primes.

#### Proof:

Induction over all prime gaps until the maximally possible one each time.$\displaystyle{ \square }$

### Foundation theorem

Only the postulation of the axiom of foundation that every nonempty subset $\displaystyle{ X \subseteq Y }$ contains an element $\displaystyle{ x_0 }$ such that $\displaystyle{ X }$ und $\displaystyle{ x_0 }$ are disjoint guarantees cycle freedom.

#### Proof:

Set $\displaystyle{ X := \{x_m : x_0 := \{\emptyset\}, x_{\omega} := \{x_1\} }$ and $\displaystyle{ x_{\acute{n}} := \{x_n\} }$ for $\displaystyle{ m \in {}^{\omega}\mathbb{N} }$ and $\displaystyle{ n \in {}^{\omega}\mathbb{N}_{\ge 2}\} }$ .$\displaystyle{ \square }$