Embedding by sections of a power of coprime degree

For $L$ ample on $S$ and $a>0$ there are $b>0$ coprime to $a$ and sections of $L^b$ indexed by $\operatorname{Option}(\operatorname{Fin}N)$ with iso-opens covering $S$ defining a closed immersion.

Notation. The iso-open of $s$ (SectionOpens.isoOpen s) is the union of the opens on which the morphism $s$ restricts to an isomorphism. sectionsMorphism k s hs is the morphism from $X$ to $\operatorname{Proj}$ of the graded polynomial ring in variables indexed by the sections, defined by sections whose iso-opens cover $X$, with $k$ the ring map from the base into $\Gamma(X,\top)$. IsAmple means that for each point $x$ and open $V\ni x$, some positive power has a global section with nonvanishing open containing $x$, inside $V$, and affine. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.