The other tutor did not prove it.

We assume that the distributive, identity and inverse principles hold.

First we must prove that (-1)(0) = 0  

[You might already accept that anything times 0 is 0, 
but even that must be proved!] 

Start with: 

     (-1)(0)

Since 0 = 0 + 0 we can replace the 0 by (0 + 0):

     (-1)(0) = (-1)(0 + 0)

We use the distributive principle on the right side:

     (-1)(0) = (-1)(0) + (-1)(0)

We subtract (-1)(0) from both sides:

           0 = (-1)(0) 

So we have proved that (-1)(0) = 0:

--------------------------------------

Now start with 

(-1)(-1) + (-1)

Since (-1) = (-1)(1), we can substitute (-1)(1) for the second term (-1),
and the above becomes:

(-1)(-1) + (-1) = (-1)(-1) + (-1)(1)

We factor (-1) out of both terms:

(-1)(-1) + (-1) = (-1)(-1 + 1)

We know that (-1 + 1) = 0 so we have

(-1)(-1) + (-1) = (-1)(0)

and we have proved above that (-1)(0) = 0, so we have proved

(-1)(-1) + (-1) = 0

and we know that 1 + (-1) = 0, so we replace the 0 on the right
by 1 + (-1)

(-1)(-1) + (-1) = 1 + (-1)

We subtract (-1) from both sides:

       (-1)(-1) = 1

Edwin