Grothendieck's Schemes in Algebraic Geometry

We formalize mainstream structures in algebraic geometry [1, 2] culminating in Grothendieck’s schemes: presheaves of rings, sheaves of rings, ringed spaces, locally ringed spaces, affine schemes and schemes. We prove that the spectrum of a ring is a locally ringed space, hence an affine scheme. Finally, we prove that any affine scheme is a scheme.