Suppose you have a triangulated region in the plane, the triangulation consisting of $n$ triangles. Take an arbitrary triangle of this triangulation and call it $\Delta_i$ with $1\leq i\leq n$.
The neighbourhood of $\Delta_i$, i.e. all the triangles around $\Delta_i$ which share a vertex or an edge with $\Delta_i$, we call this neighbourhood $N_i$.
If we can prove that you can label the vertices of $\Delta_i$ and its neighbourhood $N_i$ with just 4 colours, i.e. such that all adjacent vertices have different colour: does the 4-Colour-Theorem follow?
My thinking is: yes it follows, because $\Delta_i$ was arbitrarily chosen in this triangulation. Therefore we can conclude the whole triangulation can be 4-coloured. And if 4 colours suffice for an arbitrary triangulation, then 4 colours also suffice for any plane graph.
Would this be a valid proof strategy for the 4-Colour-Theorem, or am I taking the wrong conclusion?