Group Theory For an algebraic system, how to prove the following?

Question

I'm trying to prove the below equation

(From Elements of discrete mathematics, second edition by C. L. Liu Question 11.13)

Let $(A, +)$ be an algebraic system such that for all $a, b$ in $A$ we have

$$(a + b) + a = a$$ $$(a + b) + b = (b + a) + a$$

(1) Show that $a + (a + b) = a + b$ for all $a$ and $b$

(2) Show that $а+a = (а+b) + (а+b)$ for all $a$ and $b$.

I got the 1st part as follows:

$(a+b)+a = a$

let $a+b = c$

Hence $c+a = a$

Now LHS of statement 1 is $a+(a+b)$.

Using a from (1) we get $c+a+c$

which is equivalent to $(c+a)+c $

c

a+b= RHS

How to prove the 2nd

Answer 1

$$a+(a+b)=((a+b)+a)+(a+b)=a+b.$$ $$(a+b)+(a+b)=(a+(a+b))+(a+b)=((a+b)+a)+a=a+a.$$