Prove these definitions are equivalent:
Definition $\,(1)\,$: $A\subset X$ is not connected if for open $U, V\subset X\,\,, $$\,\,U\cap\bar{V}=\emptyset\,\,$, $\,\,\bar{U}\cap V=\emptyset\,\,$, $\,\,U\cap A\neq\emptyset\,\,$, $\,\,V\cap A\neq\emptyset\,\,$ and $\,\,A \subset U\cup V$.
-and-Definition $\,(2)\,$ $A$ is not connected if $U\cap A\neq\emptyset$,$V\cap A\neq\emptyset$, $(U\cap A)\cap (V\cap A)=\emptyset$ but $(U\cap A)\cup (V\cap A)=A$.
Attempt: For $A\subset X$ and open $U,V\subset X$ s.t. $U,V$ disconnect $A$,
(1) $(U\cap A)\cap (V\cap A)=\emptyset\implies$ $U\cap\bar{V}=\emptyset$,$\bar{U}\cap V=\emptyset$
(2) $(U\cap A)\cup (V\cap A)=A\implies$ $U\cup V=A$.