We use only the usual field axioms for the real numbers. First we prove an intermediate result.
Subtract 
from each side to get 
. Now we are ready for the final kill.
Add 
to each side to get 
.
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We use only the usual field axioms for the real numbers. First we prove an intermediate result.
Subtract from each side to get . Now we are ready for the final kill.
Add to each side to get .
The Law of Signs 
isn't normally assumed as an axiom. Rather, it is derived as a consequence of more fundamental Ring axioms 
[esp. the distributive law 
], laws which abstract the common algebraic structure shared by familiar number systems. Below are a few ways to prove the law of signs (notice that the over/underlined terms 
$\begin{eqnarray}\rm{\bf Law\ of\ Signs}\ proof\!:\ &&\rm (-x)(-y) = (-x)(-y) + \underline{x(-y + y)} = \overline{(-x+x)(-y)} + xy = xy\\ \\ \rm Equivalently,\ evaluate &&\rm\overline{(-x)(-y)\! +} \overline{ \underline {x(-y)}} \underline{ +xy_{\phantom{.}}}\ \ \text{in two ways, over or underlined first}\\ \\ \rm More\ conceptually:\quad\, &&\rm (-x)(-y)\quad\ and\ \quad xy\ \ \ \text{are both inverses of} \ \ x(-y)\\ && \text{hence they are equal by } {\bf uniqueness\ of\ inverses}\end{eqnarray}$
Indeed, the above are special cases of an analogous proof of uniqueness of additive inverses
Notice that the proofs use only ring laws (most notably the distributive law), so the law of signs holds true in every ring. The distributive law is at the foundation of every ring theorem that is nondegenerate, i.e. involves both addition and multiplication, since it is the only ring law that connects the additive and multiplicative structures that, combined, form the ring structure. Without the distributive law a ring would be far less interesting algebraically, reducing to a set with additive and multiplicative structure, but without any hypothesized relation between the two. Therefore, in a certain sense, the distributive law is the keystone of the ring structure.
well the 1 + 1 = 2 thread seemed to go down a storm!
You are thinking of the Principia Mathematica, written by Alfred North Whitehead and Bertrand Russell. Here is a relevant excerpt:
As you can see, it ends with "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." The theorem above, $\ast54\cdot43$, is already a couple of hundred pages into the book (Wikipedia says 370 or so)
I wrote a blog article a few years ago that discusses this in some detail. You may want to skip the stuff at the beginning about the historical context of Principia Mathematica. But the main point of the article is to explain the theorem above.
The article explains the idiosyncratic and mostly obsolete notation that Principia Mathematica uses, and how the proof works. The theorem here is essentially that if $\alpha$ and $\beta$ are disjoint sets with exactly one element each, then their union has exactly two elements. This is established based on very slightly simpler theorems, for example that if $\alpha$ is the set that contains $x$ and nothing else, and $\beta$ is the set that contains $y$ and nothing else, then $\alpha \cup \beta$ contains two elements if and only if $x\ne y$.
The main reason that it takes so long to get to $1+1=2$ is that Principia Mathematica starts from almost nothing, and works its way up in very tiny, incremental steps. The work of G. Peano shows that it's not hard to produce a useful set of axioms that can prove 1+1=2 much more easily than Whitehead and Russell do.
The later theorem alluded to, that $1+1=2$, appears in section $\ast110$:
I suspect you are referring to the Principia Mathematica. I direct you to a quotation from Wikipedia about how the proof doesn't appear until page 379.
I'd do it directly as follows:
$$(-1)(-1)+(-1)\stackrel{Dist.}=(-1)\left[(-1)+1\right]\stackrel{\text{additive inv.}}=(-1)\cdot 0\stackrel{text}=0$$
Thus, $\;-1\;$ is the additive inverse of $\;(-1)(-1)\;$ , but also $\;-(-1)=1\;$ is the additive inverse of $\;(-1)\;$, so by uniqueness of the inverses we're done.
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