A posterior probability combines a likelihood with its prior and normalizes over all hypotheses. I want the notation, the mechanism, and the failure mode visible at the same time.
Set-up
A probability space \((\Omega,\mathcal F,\mathbf P)\) separates outcomes, measurable events, and their probabilities. A random variable \(X:\Omega\to\mathbf R\) must be measurable.
The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.
The calculation
Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.
What survives abstraction
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
The boundary
Conditioning on a probability-zero event cannot be done by naïvely dividing. Conditional densities and regular conditional probabilities require additional structure.
I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.