capital punishment - Why does the Gemara go through each death penalty?

The Gemara Sanhedrin 50a writes the order of the death penalties.

It starts with Stoning, proving that Stoning is more severe than any other form of death penalty. Then it moves to Burning, proving that it's more severe than any other form of death penalty. Then it moves to Decapitation ...

Why?

All you have to prove is that Decapitation is stricter than Strangulation, Burning is stricter than Decapitation, and Stoning is stricter than Burning, we'd know the order by a simple syllogism.