Question: As an exercise I try to prove this inequality principle : the sense of an inequality is reversed if each side is multiplied by the same negative number. 

In steps $(2)$ and $(4)$ of the proof ( below) , I need to use the fact that : 

> 
> $$(a) (-1) = -a$$
> 
> 
> 

So I've asked myself the question : how to prove this fact ? 

The number : $-a$ is defined as the opposite (or additive inverse) of the number $a$ . Opposites ( additive inverses) are not defined in terms of multiplication. So, though basic, this fact is not absolutely trivial. 

It seems to me that I cannot use here the " rule of signs" since the sign of the number a is undetermined. 

Which *axioms* or *basic principles* could be used in an algebraic system to prove this elementary fact? 

Below, my attempt of proof regarding inequalities : 

[![enter image description here](https://i.stack.imgur.com/wbSoj.jpg)](https://i.stack.imgur.com/wbSoj.jpg)

Answer: 
You can use the fact the\begin{align}a+(-1)a&=1\times a+(-1)\times a\\&=\bigl(1+(-1)\bigr)a\\&=0\times a\\&=0.\end{align}Of course, I am using here the fact that $0\times a=0$, which can be proved as follows:\begin{align}0&=0\times a-0\times a\\&=(0+0)\times a-0\times a\\&=(0\times a+0\times a)-0\times a\\&=0\times a+(0\times a-0\times a)\\&=0\times a+0\\&=0\times a\end{align}
Using distributivity of multiplication over addition,

$$a(-1)+a=a(-1)+a\cdot1=a(1-1)=a\cdot0=0$$ so that $$a(-1)=-a.$$