A semialgebra $\mathcal{S}$ is a nonempty family of a set $X$ satisfying the properties
$1.$ $E,F\in\mathcal{S}\implies E \cap F\in \mathcal{S}$
$2.$ If $E\in\mathcal{S}$ then there exists pairwise disjoint sets $F_1,\dots, F_n\in\mathcal{S}$ such that $$X\setminus E =\bigcup_{k=1}^n F_k$$
Now, let $A,B\in\mathcal{S}$. I must prove that exists a pairwise disjoint sets $C_1,\dots, C_n\in\mathcal{S}$ such that $$A\setminus B=\bigcup _{k=1}^n C_k.$$
My attempt. We observe that $$A\setminus B=A\cap (X\setminus B)$$ then from $(2)$ of definition exists pairwise disjoint sets $B_1,\dots, B_n\in\mathcal{S}$ such that $$X\setminus B=\bigcup_{k=1}^n B_k,$$ then $$A\setminus B=\bigcup_{k=1}^n(A\cap B_k).$$
If define $C_k=A\cap B_k$ for $k=1,\dots, n$ we finished. This is correct?