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$(1-2)(1-2) =1-2-2+4=1$, using the distributive property. If it's not valid, why?

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    $\begingroup$ Sure, but it's hard to imagine a situation in which you would know $-2×-2=4$ without already having proved $-1×-1=1$. $\endgroup$ – MJD Dec 21 '16 at 3:02
  • $\begingroup$ How do you know $(-2)(-2) = 4$? I think you need to know $(-1)(-1) = 1$ to conclude this. $\endgroup$ – André 3000 Dec 21 '16 at 3:02
  • $\begingroup$ Hint for a correct proof. Prove that $(-1)\cdot x = (-x)$ Prove then that $(-1)\cdot (-1)=-(-1)=1$ due to the uniqueness of inverses. $\endgroup$ – JMoravitz Dec 21 '16 at 3:09
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It's not valid unless you have already proved that $(-2)(-2)=4$.

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Here's a variation though that does work. Suppose we agree that $(-1)(-1)=x$, and we just don't know what $x$ is. Then $$0=0\cdot 0=(1-1)\cdot(1-1)=1-1-1+(-1)(-1)=-1+x$$ So, $-1+x=0$, and thus the only thing $x$ can be is $1$.

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