Occasionally, I see arithmetic/algebraic operations performed in the context of equations or inequalities that include the $\Pr()$ function. $\Pr()$ is an unusual animal, because it can live within an equation or inequality, and its input is another equation or inequality. For example, $$\Pr(X < 2) = \frac{3}{5},$$$$\Pr(X > 4) = \Pr(X < 2) + \frac{1}{2},$$$$1 < \Pr(\Pr(X = 5) \le X).$$ When these start being manipulated, i.e., to normalize random variables, I'm a bit unclear on what rules apply. For example, in the second equation, to multiply both sides of $X < 2$ by $10$, do we also need to multiple both sides of $X > 4$ by $10$? Do we also multiply $\frac{1}{2}$ by $10$? If we multiply both sides of the third equation by $2$, do we distribute the $2$ to the innermost equation, $X = 5$? What about squaring both sides, etc.?
What would be a simple list of rules that explains how to manipulate these equations in the way ordinary algebraic equations are manipulated at a high school level?