For the longest time I thought this was something you had to assume using the logical negation of truth tables, but you can prove this using the field axioms instead:
1 * -1 = -1
-1 = -1
1 -1 -1 = 0 + (-1) = -1
(-1)(1 -1 -1) = (-1)(-1)
distributing across brackets:
1 * -1 + (-1)(-1) + (-1)(-1) = (-1)(-1)
-1 + (-1)(-1) + (-1)(-1) - (-1)(-1) = (-1)(-1) - (-1)(-1)
-1 + (-1)(-1) + 0  = 0
-1 + (-1)(-1)  = 0
1 -1 + (-1)(-1)  = 0 + 1 = 1
0 + (-1)(-1)  = 1
(-1)(-1)  = +1

negative one times negative one equals positive one

same