how to prove original intervals and canonical form of intervals have the same interval graph
You haven't described how a system of intervals are converted to canonical form, but here is one way to do it. We find an order-preserving mapping (satisfies $f(x) &lt; f(y)$ iff $x &lt; y$) from the coordinates in the original system to the integers $\{1,\ldots,2n\}$ (we do this by sorting all of the original coordinates). We get the same interval graph since the mapping is order-preserving – that's a nice exercise.