Understanding Lang's application of two-minus sign products This is very basic stuff from the first chapter of Lang's Basic Mathematics, but I can't get past it. Lang here uses a previously established rule to prove that (-1)(-1) = 1

but I cannot reason out his steps. How is what he's worked out an application of the rule he cites? What's a and what's b here? I tried doing it on my own and couldn't reason through it.
 A: Welcome to MSE! The part that might be a bit confusing is that on each of the $2$ equal signs in the steps he's taking $a$ and $b$ to be different values. I'll try to make it a bit clearer.

We have the rule
$$
(-\color{green}{a})\color{blue}{b} =-(\color{green}{a}\color{blue}{b})
$$
So taking $\color{green}{a} =\color{green}{1}$ and $\color{blue}{b} = \color{blue}{-1}$ we get
$$
(-\color{green}{1})(\color{blue}{-1}) =-(\color{green}{1}(\color{blue}{-1})) \tag{1}
$$
Now, we also have the second property
$$
\color{purple}{a}(-\color{darkblue}{b}) = -(\color{purple}{a}\color{darkblue}{b})
$$
So now, taking $\color{purple}{a} = \color{purple}{1}$ and $\color{darkblue}{b} = \color{darkblue}{1}$ we get
$$
\color{purple}{1}(-\color{darkblue}{1}) = -(\color{purple}{1}\cdot\color{darkblue}{1}) = - 1 \tag{2}
$$
since $1\cdot 1 = 1$ by definition of the multiplicative identity. Now notice that the left-hand side of $(2)$ is inside the outermost parenthesis in the right-hand side of $(1)$ so combining $(1)$ and $(2)$ gives
$$
(-1)(-1) \overset{(1)}{=} -(\color{brown}{1(-1)}) \overset{\color{brown}{(2)}}{=} -(\color{brown}{-1})
$$
Lastly, the author uses the previously proven result $-(-1) = 1$ to get the last equality.
A: The notation is a little slippery, and you may be interested to hear that historically, people squabbled for centuries over the definition of negative numbers and how to multiply them together.
Let's add a useless 0 to the equation, then try to work out what
$$
0+(-1)\times(-1)
$$
is, is it $-1$ or $1$ or $0$ or something else?
Applying the rule once gives us
$$
0+-(1)\times(-1)
$$
which is
$$
0-(1)\times(-1)
$$
by the usual rules for adding negative numbers.
Then applying the rule again gives
$$
0-(-(1\times1))
$$
Now clearly $1\times1=1$ so this is
$$
0-(-1)
$$
and using the rule for subtracting negative numbers gives us
$$
0+1=1
$$
so we have $(-1)\times(-1)=1$.
