A simple connected planar graph with at least three vertices has at most three n minus six edges. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
The mathematical object
A planar embedding divides the sphere into vertices, edges, and faces. Euler's relation \(|V|-|E|+|F|=2\) constrains density, while the dual \(G^\ast\) records adjacency of faces.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
One explicit computation
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
Why the identity matters
A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.
Where it can fail
Planarity is a property of a graph, while a plane graph includes a chosen embedding. The dual depends on that embedding.
The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.