In a graph $\mathcal G=(V, E)$ let $a,z \in V$
For proving the shortest distance from $a$ to $z$:
- We have a sub path $i$ to $j$ such that $a\rightarrow i \rightarrow j \rightarrow z$ If we find a smaller.
- Sub path $i'\rightarrow j'$ such that $ai'j'z < aijz$.
- We say that to prove $a\rightarrow z$ is the shortest path we have to prove it's sub-path has a shortest path.