Four points on the projective line determine a cross ratio invariant under fractional linear transformations. The example is deliberately concrete; it is a test of the statement, not a substitute for it.
Start locally
A Lie group \(G\) is simultaneously a smooth manifold and a group. Hyperbolic metrics on \(\mathbf H\) and \(\mathbf D\) have Lie groups of Möbius isometries.
A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.
Compute before generalising
A worked instance is useful here because it exposes every index that the compressed statement hides.
The global view
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.
Edge conditions
Euclidean drawings distort hyperbolic distance and angle behavior unless the chosen model is conformal. Boundary points are not interior points at finite distance.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.