In my Vector Calculus course, the professor is rigorous enough that we do a decent number of proofs, but not rigorous enough to go all the way with manifolds/differential forms/etc. One proof in particular that didn't sit completely well with me was our proof of Stokes' Theorem using Green's Theorem.
Now, it's my understanding that Green's Theorem is a specific case of Stokes' Theorem (which are both specifics of the generalized Stokes' Theorem), and so it feels to me like we're sort of assuming the conclusion when we use it in the proof. Here's the outline of the proof, if it matters.
The basic idea was to show that given a parametrization of a surface in $\mathbb{R}^3$, the boundary of the surface is parameterized by the boundary of the parameterization. And so you go from a line integral around the boundary of the surface to a line integral around the boundary of the parametrization. From there you apply Green's theorem to take the double integral over the interior of the region, and then do the parametrization in reverse to get back the original surface.
Best Answer
Good question. It is fairly common to prove a theorem by showing that the general situation can be reduced to a particular case. That's exactly what the argument you mention does.
There is no contradiction. Green is a particular case of Stokes, but Stokes (at least the version you were shown) can be proven using Green.