$\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\jleq}[1]{\justifyed{#1}{\leq}} \newcommand{\jeqtxt}[1]{\jeq{\text{#1}}} \newcommand{\jl}[1]{\justifyed{#1}{<}} \newcommand{\justifyed}[2]{\stackrel{#1}{#2}} \newcommand{\jeq}[1]{\justifyed{#1}{=}} \newcommand{\jleqref}[1]{\jleqtxt{\ref{#1}}} \newcommand{\jeqref}[1]{\jeqtxt{\ref{#1}}} \newcommand{\jleqtxt}[1]{\jleq{\text{#1}}} \newcommand{\df}{\mathrel{\mathop:}=} \newcommand{\norm}[1]{\left|#1\right|}$ Isosceles triangle principle: Let $\norm{\cdot}$ be a non-Archimedean norm on a field $\mathbb{F}$. Then $\forall\ x,y\in\mathbb{F}$ with $\norm{x}<\norm{y}$ it holds that $\norm{x+y}=\norm{y}$.
Proof: Let $x,y\in\mathbb{F}$ with $\norm{x}<\norm{y}$. We now calculate \begin{align} \norm{x-y}\jleqtxt{non-Arch.}\max\set{\norm{x},\norm{y}}=\norm{y}=\norm{x-(x-y)}\jleqtxt{non-Arch.}\max\set{\norm{x},\norm{x-y}}. \end{align} From this equation we get that $\norm{y}\leq\max\set{\norm{x},\norm{x-y}}$ but since we assumed $\norm{x}<\norm{y}$, we have that $\norm{y}\nleq\norm{x}$ so $\max\set{\norm{x},\norm{x-y}}=\norm{x-y}$ actually. Together with the equation above, we get that $\norm{x-y}=\norm{y}$.
Remark: The name "Isosceles Triangle Principle" comes from the geometric interpretation: If $x$ and $y$ are two points of a triangle which has the origin $0$ as it's third point, the sides have the length $x$, $y$ and $x-y$. The principle then says that the third side is as long as the longer one of the others, so every "triangle" is isosceles with respect to a non-Archimedean norm.
Lemma: Let $\norm{\cdot}: \mathbb{F}\rightarrow\mathbb{R}_{\geq 0}$ be a non-Archimedean norm and let $\forall\ i\in \set{0, \ldots, n}: a_i\in\mathbb{F}$. If $\forall\ 0\norm{a_i}$ it holds that $\norm{\sum_{i=0}^n a_i}=\norm{a_0}$.
Proof: We prove this statement by induction. For $n=0$, the statement is trivial. Now assume that the statement is proven for $n$ and let $a_{n+1}\in\mathbb{F}$ with $\norm{a_0}>\norm{a_{n+1}}$. Now set $x\df a_{n+1}$ and $y=\sum_{i=0}^n a_i$ and note that: $$ \norm{y}\jeqtxt{I.H.}\norm{a_0}>\norm{a_{n+1}}=\norm{x} $$ and we can apply the Isosceles Triangle Principle (ITP): $$ \norm{\sum_{i=0}^{n+1} a_i}=\norm{\sum_{i=0}^{n} a_i + a_{n+1}} = \norm{x+y}\jeqtxt{ITP}\norm{y}=\norm{\sum_{i=0}^n a_i}\jeqtxt{I.H.}\norm{a_0}.$$