Exponential function properties

I am learning about exponential function so I need some help to understand.

I am learning about properties of function $\displaystyle f(x) = 2^x ,x \in Q$

I need explanation of one property.

Property is: for all $\displaystyle x_1 ,x_2 \in Q,x_1 < x_2 \Rightarrow 2^{x_1 } < 2^{x_2 } $

Proof of this property is:

let $\displaystyle x_1 = \frac{{p_1 }}{{q_1 }},x_2 = \frac{{p_2 }}{{q_2 }},p_1 ,p_2 \in Z,q_1 ,q_2 \in N$

let $\displaystyle n$ be common factor of numbers $\displaystyle q_1,q_2$ so we have $\displaystyle x_1 = \frac{{m_1 }}{n},x_2 = \frac{{m_2 }}{n}$...

I need explanation how did $\displaystyle x_1 = \frac{{p_1 }}{{q_1 }},x_2 = \frac{{p_2 }}{{q_2 }}$ become $\displaystyle x_1 = \frac{{m_1 }}{n},x_2 = \frac{{m_2 }}{n}$?