The triangle inequality

Given: An ordered field $F$ and elements $x,y\in F$.

For every $u\in F$, $-\mathrm{\mid }u\mathrm{\mid }\le u\le \mathrm{\mid }u\mathrm{\mid }$, and $\mathrm{\mid }u\mathrm{\mid }$ equals $u$ or $-u$ (Basic properties of the absolute value).

Order compatible with addition: if $a\le b$ and $c\le d$, then $a+c\le b+d$. Order is preserved by adding a constant and by adding inequalities states the STRICT forms and only those ($a<b\Rightarrow a+c<b+c$, and $a<b$ with $c<d$ giving $a+c<b+d$); the nonstrict form used here is those two together with the cases $a=b$ and $c=d$, settled by trichotomy, the order being total (Ordered field). Explicitly: if $a<b$ and $c<d$ the second strict form applies; if $a<b$ and $c=d$ the first gives $a+c<b+c=b+d$; if $a=b$ and $c<d$ the first gives $a+c<a+d=b+d$; and if $a=b$ and $c=d$ the two sides are equal.

Field and order arithmetic: $-(a+b)=-a+(-b)$, and $a\le b⟺-b\le -a$ (Ordered field).