If \(A\times B \subseteq C\times D\) and \(A\times B \neq \Phi\), Prove That \(A \subseteq C\) and \(B \subseteq D\)

Since \[ A\times B \neq \Phi, \] there exist \[ a\in A,\quad b\in B. \]

Let \[ x\in A. \]

Since \[ b\in B, \] we have \[ (x,b)\in A\times B. \]

But \[ A\times B \subseteq C\times D, \] therefore \[ (x,b)\in C\times D. \]

Hence, \[ x\in C. \]

Thus, \[ A\subseteq C. \]

Now let \[ y\in B. \]

Since \[ a\in A, \] we have \[ (a,y)\in A\times B. \]

Again, \[ A\times B \subseteq C\times D, \] therefore \[ (a,y)\in C\times D. \]

Hence, \[ y\in D. \]

Thus, \[ B\subseteq D. \]

Therefore,

\[ \boxed{ A\subseteq C \quad \text{and} \quad B\subseteq D } \]