Is this proof optimized well enough?

Question

Suppose a,b,c,d € R. Then If a+c=b+c, then a=b ( this is the question)

My answer; Assume a+c = b+c then

a+c+(-c)=(-c)+b+c by (Existence of Additive Inverses)

Thus

a+(c+(-c)=((-c)+c)+b by (Associativity of Addition)

Since (c+(-c))=0 by (Existence of Additive Inverses)

a+(0)=(0)+b

which is

a=b.

Answer 1

It looks correct, but I'd rather write $a+c+(-c)=b+c+(-c)$ than what you wrote just in case it is not commutative the sum where you are working (very strage but it can happen)