= Solution
An algebraic way to prove the <set identity> is to show that $D=\overline A\cup\overline B$ is a <complement of a set> for $A\cap B$. By the <distributive property>,
$$
(A\cap B)\cap D=(A\cap B\cap\overline A)\cup(A\cap B\cap\overline B)=\varnothing,
$$
and
$$
(A\cap B)\cup D=(A\cup D)\cap(B\cup D)=\mathcal U\cap\mathcal U=\mathcal U.
$$
A <complement of a set> is unique: if $D,E$ both complement $S$, then $D=D\cap(S\cup E)=D\cap E$, and similarly $E=D\cap E$. Consequently
$$
\boxed{\overline{A\cap B}=\overline A\cup\overline B.}
$$
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