We know that every rational number can be expressed in the form of a ratio of two integers. Using this, Lets label the first rational number as $p_1$ and the second rational number as $p_2$. By the theorem we mentioned above, we can express $p_1$ as $p_1 = \frac{x_1}{y_1}$ where $x_1,y_1$ are integers. Similarly we can express $p_2 = \frac{x_2}{y_2}$ where $x_2,y_2$ are integers. Multiplying $p_1,p_2$ we get $p_1\cdot p_2 = \frac{x_1x_2}{y_1y_2}$. But multiplication of two integers is always an integer and hence $\frac{x_1x_2}{y_1y_2}$ is a rational number. Therefore, multiplication of any two rational numbers is a rational number.