Formal proof for $(-1) \times (-1) = 1$ Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?

We use only the usual field axioms for the real numbers. First we prove an intermediate result.
$0\times 0$
$=0\times(0+0)$
$=0\times 0+0\times 0$
Subtract $0\times 0$ from each side to get $0=0\times 0$. Now we are ready for the final kill.
$0$
$=0\times 0$
$=(1-1)\times(1-1)$
$=1\times 1+1\times (-1)+(-1)\times 1+(-1)\times (-1)$
$=1+(-1)+(-1)+(-1)\times (-1)$
$=(-1)+(-1)\times (-1)$
Add $1$ to each side to get $1=(-1)\times (-1)$.
