Given the following axiomatic system :

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

We want to prove:

For all A : A+(-A) = 0

Proof:

From axiom (4) if we put A = -B we have : (-B)+B =0 => B+(-B) =0 ,by axiom (1)

Hence by changing the variables we have :

For all A : A+(-A) = 0

I s that proof correct??