I need to prove this: a,b,c ∈ and c ∉ 0 and if ac=bc then a=b

I am only allowed to use the axioms for

I have ac-bc=0

prove: ac-bc=c(a+(-b)) (I used the axiom associative)

because c ∉ 0 then (a+(-b))=0 so a=b

Can you tell if my pove is right and which axioms I need to use to prove this?

thank you