I beg your pardon! Steps 1-4 are exactly a direct proof that -1 x +1 = -1. A is defined as -1, not zero. Therefore since A is non-zero, A/A by definition is equal to +1. Therefore -1/-1 is also equal to +1. Carrying the divisor to the right via multiplication yields the completion of the proof. Step 5 then shows that the conclusion which was proven true for value i remains true for value i+1 and even for i=n (inductive reasoning).

Additionally I confirmed the proof via the methodology stated beneath #5. If you still doubt that this is a valid math proof, see a math professor and show it to him/her. I've had to provide many a mathematical proof back in my college days, so I know what it takes to make one.

BTW, your 0/0 = 1 yielding 0 = 1 would be incorrect on two counts. First, it is improper to divide by zero or infinity. Second, since dividing by 0 isn't permitted, you should have used multiplication to conclude that 0 = 0 x 1; and since 0 x 1 = 0, you should have reasoned that 0 = 0. 

"We have seen the enemy, and he is us!" (POGO)
Edited 3 times by dg May 16 11 10:30 AM.