Improve statement lemma Tag 00JO

Thanks to abcxyz

algebra.tex

@@ -11974,15 +11974,16 @@ \section{Proj of a graded ring}
 
 \begin{lemma}
 \label{lemma-Z-graded}
-Let $S$ be a $\mathbf{Z}$-graded ring.
-Let $f \in S_d$, $d > 0$ and assume $f$ is invertible in $S$.
-The set $G \subset \Spec(S)$ of $\mathbf{Z}$-graded primes of $S$
+Let $S$ be a $\mathbf{Z}$-graded ring containing a homogeneous
+invertible element of positive degree. Then the set
+$G \subset \Spec(S)$ of $\mathbf{Z}$-graded primes of $S$
 (with induced topology) maps homeomorphically to $\Spec(S_0)$.
 \end{lemma}
 
 \begin{proof}
 First we show that the map is a bijection by constructing an inverse.
-Namely, if $\mathfrak p_0$ is a prime of $S_0$, then $\mathfrak p_0S$
+Let $f \in S_d$, $d > 0$ be invertible in $S$.
+If $\mathfrak p_0$ is a prime of $S_0$, then $\mathfrak p_0S$
 is a $\mathbf{Z}$-graded ideal of $S$ such that
 $\mathfrak p_0S \cap S_0 = \mathfrak p_0$. And if $ab \in \mathfrak p_0S$
 with $a$, $b$ homogeneous, then