Theorems, lemmas, and other proclamations

Theorems and lemmas are varieties of theorem environments. In this document, a theorem environment called lemma has been created, which is used below. Also, there is a proof, which is in the predefined pf environment. The lemma and proof below illustrate the use of the enumerate environment.

Lemma 2.1   Let f, g∈A(X) and let E, F be cozero sets in X.
  1. If f is E-regular and F⊆E, then f is F-regular.

  2. If f is E-regular and F-regular, then f is E∪F-regular.

  3. If f (x)≥c > 0 for all x∈E, then f is E-regular.


\begin{pf} \begin{enumerate} \par \item Obvious. \par \item Let $h, k\in A(X)$\ ... ... C^*(X)\subseteq A(X)$, and $h^{-1} f\vert _E=1$. \par \end{enumerate}\end{pf}

Another theorem-type environment was defined at the beginning of this document, called definition. Here is an example of it:

Definition 2.1   For f∈A(X), we define

$\displaystyle \cal {Z}$(f )= {E∈Z[X] f is Ec-regular}. (2.1)