Given the following axioms:


For all A,B,C:


1) A+B= B+A .................................................. ......AB= BA


2) (A+B)+C= A+(B+C)........................................... ....(AB)C= A(BC)


3) A+0 =A................................................ .................1A=A


4) For all A ,there exists B : A+B=0.....................................For all A =/=0 ,there exists B: AB=1


5) A(B+C)= AB+AC


6) 1=/=0

And the theorems:

7) 0A = 0

8) AB=0 <=> A=0 v B=0

9) A+C=B+C => A=B

Prove : AA =BB => A+B=0 v A=B