Can you check next statements, and they are proofs?
Statement 1. Lets $A, A_1, A_2$ - are abelian groups and $$A = A_1\oplus A_2.$$ Then $$A/A_1=A_2.$$ Proof: $$A=\{(a_1, a_2)|a_1\in A_1,~~a_2\in A_2\}.$$ $$x = (x_1, x_2)\sim y = (y_1, y_2)\Leftrightarrow x-y\in A_1 \Leftrightarrow x_2=y_2.$$ So, homomorphism $\varphi : A/A_1\to A_2$, such that $$\varphi(a_1, a_2) = a_2,$$ is isomorphism. $\blacksquare$
Statement 2. Lets $A\supset B$ - abelian, then $$A = B\oplus A/B$$
Proof: $A\supset B$, therefore $$\exists C\subset A: A=B\oplus C.$$ And from first statement: $$C = A/B.$$ $\blacksquare$
Thanks.