We know how to count: 1, 2, 3, 4, etc.
For every $a$, $b$, $c$ $\in\mathbb{R}$:
$(a+b)+c=a+(b+c)$.
$(ab)c=a(bc)$.
$a+b=b+a$.
$ab=ba$.
$a(b+c)=ab+ac$.
There exists $\mathit{0}$ such that for every $a\in\mathbb{R}$:
$a+\textit{0}=a$.
There exists $\mathit{1}$ such that for every $a\in \mathbb{R}$:
$\textit{1}a=a$.
For every $x\in\mathbb{R}$ there exists $y\in\mathbb{R}$ such that:
$x+y=\mathit{0}$.
For every $x \in\mathbb{R}^{\times}$ there exists $y\in \mathbb{R}^{\times}$ such that:
$xy=\mathit{1}$.
$\textit{0}\not\in\mathbb{R}^{+}$
For every $a\in\mathbb{R}$ either $a\in\mathbb{R}^{+}$ or $-a\in\mathbb{R}^{+}$, but not both.
For every $a,b\in\mathbb{R}^{+}$:
$a+b\in\mathbb{R}^{+}$.
$ab\in\mathbb{R}^{+}$.
For $a_{n}$, $a_{k} \in\mathbb{R}$ and $N, M$ counting numbers.
As definitions: $\sum_{n=1}^{0}a_{n}=\textit{0}$, $\sum_{n=1}^{1}a_{n}=a_{1}$
and $\sum_{n=1}^{N+1}a_{n}=\left(\sum_{n=1}^{N}a_{n}\right)+a_{N+1}$.
$\prod_{n=1}^{0}a_{n}=\textit{1}$, $\prod_{n=1}^{1}a_{n}=a_{1}$
and $\prod_{n=1}^{N+1}a_{n}=\left(\prod_{n=1}^{N}a_{n}\right)a_{N+1}$.
As theorems: $\sum_{n=1}^{N+M}a_{n}=\sum_{n=1}^{N}a_{n}+\sum_{n=N+1}^{N+M}a_{n}$
$\prod_{n=1}^{N+M}a_{n}=\left(\prod_{n=1}^{N}a_{n}\right)\left(\prod_{n=N+1}^{N+M}a_{n}\right)$
For every $\sigma\in\textrm{S}_{N}$:
$\sum_{n=1}^{N}a_{\sigma(n)}=\sum_{n=1}^{N}a_{n}$.
$\prod_{n=1}^{N}a_{\sigma(n)}=\prod_{n=1}^{N}a_{n}$.
$\left(\sum_{n=1}^{N}a_{n}\right)\left(\sum_{k=1}^{M}b_{k}\right)=\sum_{n, k}a_{n}b_{k}$.
Thread to start your mathematical journey from scratch.