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authorNorbert Preining <norbert@preining.info>2021-05-27 03:01:40 +0000
committerNorbert Preining <norbert@preining.info>2021-05-27 03:01:40 +0000
commit5f1cfa686748068fedb95d8dafda3dc7f999c948 (patch)
tree874fa8430be12fe4633924f3b707428170316a46 /macros/latex/contrib/beamer
parent8982c52e00ffb3fb5a913fc60913ebd9070f091a (diff)
CTAN sync 202105270301
Diffstat (limited to 'macros/latex/contrib/beamer')
-rw-r--r--macros/latex/contrib/beamer/CHANGELOG.md15
-rw-r--r--macros/latex/contrib/beamer/base/beamer.cls5
-rw-r--r--macros/latex/contrib/beamer/base/beamerarticle.sty2
-rw-r--r--macros/latex/contrib/beamer/base/beamerbaseauxtemplates.sty47
-rw-r--r--macros/latex/contrib/beamer/base/beamerbasecompatibility.sty2
-rw-r--r--macros/latex/contrib/beamer/base/beamerbasetoc.sty13
-rw-r--r--macros/latex/contrib/beamer/base/themes/outer/beamerouterthemeinfolines.sty11
-rw-r--r--macros/latex/contrib/beamer/doc/beamerexample-conference-talk.pdfbin324851 -> 324864 bytes
-rw-r--r--macros/latex/contrib/beamer/doc/beamerexample-lecture-beamer-version.pdfbin531872 -> 531884 bytes
-rw-r--r--macros/latex/contrib/beamer/doc/beamerexample-lecture-print-version.pdfbin425300 -> 425313 bytes
-rw-r--r--macros/latex/contrib/beamer/doc/beamerug-globalstructure.tex2
-rw-r--r--macros/latex/contrib/beamer/doc/beamerug-installation.tex2
-rw-r--r--macros/latex/contrib/beamer/doc/beamerug-localstructure.tex7
-rw-r--r--macros/latex/contrib/beamer/doc/beamerug-solutions.tex4
-rw-r--r--macros/latex/contrib/beamer/doc/beamerug-tutorial.tex2
-rw-r--r--macros/latex/contrib/beamer/doc/beamerug-workflow.tex2
-rw-r--r--macros/latex/contrib/beamer/doc/beameruserguide.pdfbin1839214 -> 1840486 bytes
-rw-r--r--macros/latex/contrib/beamer/doc/beameruserguide.tex5
18 files changed, 92 insertions, 27 deletions
diff --git a/macros/latex/contrib/beamer/CHANGELOG.md b/macros/latex/contrib/beamer/CHANGELOG.md
index 25609aee6c..72e7a2204e 100644
--- a/macros/latex/contrib/beamer/CHANGELOG.md
+++ b/macros/latex/contrib/beamer/CHANGELOG.md
@@ -8,6 +8,18 @@ a major and minor version only.
## [Unreleased]
+## [v3.63]
+
+### Changed
+
+- adding `lastsection` option to toc (#675)
+
+### Fixed
+
+- Removed superflous period in combination with the `firstsection` option (#674)
+- Behaviour of `\label` in article mode (#685)
+- Problem with date position in infolines theme (#684)
+
## [v3.62]
### Fixed
@@ -130,7 +142,8 @@ a major and minor version only.
- Support for `noxcolor` with `beamerarticle` (see #483)
- Treatment of mixed overlay/alert specifications
-[Unreleased]: https://github.com/josephwright/beamer/compare/v3.62...HEAD
+[Unreleased]: https://github.com/josephwright/beamer/compare/v3.63...HEAD
+[v3.63]: https://github.com/josephwright/beamer/compare/v3.62...v3.63
[v3.62]: https://github.com/josephwright/beamer/compare/v3.61...v3.62
[v3.61]: https://github.com/josephwright/beamer/compare/v3.60...v3.61
[v3.60]: https://github.com/josephwright/beamer/compare/v3.59...v3.60
diff --git a/macros/latex/contrib/beamer/base/beamer.cls b/macros/latex/contrib/beamer/base/beamer.cls
index 7854b07b04..135969d997 100644
--- a/macros/latex/contrib/beamer/base/beamer.cls
+++ b/macros/latex/contrib/beamer/base/beamer.cls
@@ -12,7 +12,7 @@
% See the file doc/licenses/LICENSE for more details.
\ProvidesClass{beamer}
- [2021/03/19 v3.62 A class for typesetting presentations]
+ [2021/05/26 v3.63 A class for typesetting presentations]
% Setup modes and check for article option
% Also loads packages required by both beamer and beamerarticle
@@ -74,6 +74,7 @@
\newcount\beamer@currentsubsection
\newcount\beamer@coveringdepth
\newcount\beamer@sectionadjust
+\newcount\beamer@toclastsection
\newcount\beamer@tocsectionnumber
\newif\ifbeamer@sansmath
@@ -250,7 +251,7 @@
bookmarksopen=true,%
pdfborder={0 0 0},%
pdfhighlight={/N},%
- linkbordercolor={.5 .5 .5}}{hyperref}
+ linkbordercolor=[rgb]{0.5,0.5,0.5}}{hyperref}
\DeclareOptionBeamer{pgf}{\PassOptionsToPackage{#1}{pgf}}
\DeclareOptionBeamer{hyperref}{\PassOptionsToPackage{#1}{hyperref}}
diff --git a/macros/latex/contrib/beamer/base/beamerarticle.sty b/macros/latex/contrib/beamer/base/beamerarticle.sty
index f855632e28..195e49a0e3 100644
--- a/macros/latex/contrib/beamer/base/beamerarticle.sty
+++ b/macros/latex/contrib/beamer/base/beamerarticle.sty
@@ -12,7 +12,7 @@
% See the file doc/licenses/LICENSE for more details.
\ProvidesPackage{beamerarticle}
- [2021/03/19 v3.62 beamer input in article mode]
+ [2021/05/26 v3.63 beamer input in article mode]
% Setup modes and check for article option
% Also loads packages required by both beamer and beamerarticle
diff --git a/macros/latex/contrib/beamer/base/beamerbaseauxtemplates.sty b/macros/latex/contrib/beamer/base/beamerbaseauxtemplates.sty
index 3725661462..28cfebc886 100644
--- a/macros/latex/contrib/beamer/base/beamerbaseauxtemplates.sty
+++ b/macros/latex/contrib/beamer/base/beamerbaseauxtemplates.sty
@@ -185,8 +185,16 @@
% (sub-)section in toc: sections numbered
-\defbeamertemplate{section in toc}{sections numbered}
-{\leavevmode\inserttocsectionnumber. \inserttocsection\par}
+\defbeamertemplate{section in toc}{sections numbered}{%
+ \leavevmode%
+ % prevents the period to be printed with the first/last section option
+ \ifnum\beamer@tempcount>\beamer@toclastsection
+ \else
+ \ifnum\beamer@tempcount>0
+ \inserttocsectionnumber.
+ \fi\fi%
+ \inserttocsection\par%
+}
\defbeamertemplate{subsection in toc}{sections numbered}
{\leavevmode\leftskip=2.5em\inserttocsubsection\par}
@@ -554,22 +562,43 @@
{\hskip\Gm@lmargin\hbox to\@tempdima{\textwidth=\@tempdima\hsize=\textwidth\relax#1\hfil}\hskip\Gm@rmargin}
% Framenumbers in footline
-\defbeamertemplate{page number in head/foot}{framenumber}{\insertframenumber}
+\defbeamertemplate{page number in head/foot}{framenumber}{%
+ \setbox\beamer@tempbox=\hbox{\inserttotalframenumber}%
+ \beamer@tempdim=\wd\beamer@tempbox%
+ \makebox[\beamer@tempdim][r]{\insertframenumber}%
+}
-\defbeamertemplate{page number in head/foot}{totalframenumber}{\insertframenumber\,/\,\inserttotalframenumber}
+\defbeamertemplate{page number in head/foot}{totalframenumber}{%
+ \setbox\beamer@tempbox=\hbox{\inserttotalframenumber\,/\,\inserttotalframenumber}%
+ \beamer@tempdim=\wd\beamer@tempbox%
+ \makebox[\beamer@tempdim][r]{\insertframenumber\,/\,\inserttotalframenumber}%
+}
\defbeamertemplate{page number in head/foot}{appendixframenumber}{%
+ \ifnum\insertappendixframenumber<\insertmainframenumber%
+ \setbox\beamer@tempbox=\hbox{\insertmainframenumber\,/\,\insertmainframenumber}%
+ \else%
+ \setbox\beamer@tempbox=\hbox{\insertappendixframenumber\,/\,\insertappendixframenumber}%
+ \fi%
+ \beamer@tempdim=\wd\beamer@tempbox%
\ifbeamer@inappendix%
- \insertframenumberinappendix\,/\,\insertappendixframenumber%
+ \makebox[\beamer@tempdim][r]{\insertframenumberinappendix\,/\,\insertappendixframenumber}%
\else%
- \insertframenumber\,/\,\insertmainframenumber%
+ \makebox[\beamer@tempdim][r]{\insertframenumber\,/\,\insertmainframenumber}%
\fi%
}
-\defbeamertemplate{page number in head/foot}{pagenumber}{\insertpagenumber}
-
-\defbeamertemplate{page number in head/foot}{totalpagenumber}{\insertpagenumber\,/\,\insertpresentationendpage}
+\defbeamertemplate{page number in head/foot}{pagenumber}{%
+ \setbox\beamer@tempbox=\hbox{\insertpresentationendpage}%
+ \beamer@tempdim=\wd\beamer@tempbox%
+ \makebox[\beamer@tempdim][r]{\insertpagenumber}%
+}
+\defbeamertemplate{page number in head/foot}{totalpagenumber}{%
+ \setbox\beamer@tempbox=\hbox{\insertpresentationendpage\,/\,\insertpresentationendpage}%
+ \beamer@tempdim=\wd\beamer@tempbox%
+ \makebox[\beamer@tempdim][r]{\insertpagenumber\,/\,\insertpresentationendpage}%
+}
% Footline
diff --git a/macros/latex/contrib/beamer/base/beamerbasecompatibility.sty b/macros/latex/contrib/beamer/base/beamerbasecompatibility.sty
index 75050cfcab..17ac0b45f0 100644
--- a/macros/latex/contrib/beamer/base/beamerbasecompatibility.sty
+++ b/macros/latex/contrib/beamer/base/beamerbasecompatibility.sty
@@ -107,7 +107,7 @@
<article>
{
\IfFormatAtLeastTF{2020-10-01}
- {\AddToHook{env/begin/document}[beamer]{\beamer@lastminutepatches}}
+ {\AddToHook{begindocument/end}[beamer]{\beamer@lastminutepatches}}
{%
\let\beamer@origdocument\document
\def\document{%
diff --git a/macros/latex/contrib/beamer/base/beamerbasetoc.sty b/macros/latex/contrib/beamer/base/beamerbasetoc.sty
index 9b39d707e0..0ea5318a59 100644
--- a/macros/latex/contrib/beamer/base/beamerbasetoc.sty
+++ b/macros/latex/contrib/beamer/base/beamerbasetoc.sty
@@ -63,6 +63,7 @@
\define@key{beamertoc}{part}{\beamer@showpartnumber=#1\relax}
\define@key{beamertoc}{sections}{\beamer@parsetocsections#1[]}
\define@key{beamertoc}{firstsection}{\beamer@sectionadjust=-#1\relax\advance\beamer@sectionadjust by 1\relax}
+\define@key{beamertoc}{lastsection}{\beamer@toclastsection=#1}
\def\beamer@parsetocsections{\@ifnextchar<\beamer@@parsetocsections\beamer@@@parsetocsections}
\def\beamer@@parsetocsections#1[]{\def\beamer@tocsections{#1}}
\def\beamer@@@parsetocsections#1[]{\def\beamer@tocsections{<all:#1>}}
@@ -105,6 +106,10 @@
\def\beamer@hidetext{hide}
+% initialize count with something large
+\beamer@toclastsection=\maxdimen
+\multiply\beamer@toclastsection by 2
+
\def\sectionintoc{}
\def\beamer@sectionintoc#1#2#3#4#5{%
\ifnum\c@tocdepth>0%
@@ -116,8 +121,12 @@
\expandafter\only\beamer@tocsections{\gdef\beamer@todo{%
\beamer@tempcount=#5\relax%
\advance\beamer@tempcount by\beamer@sectionadjust%
- \ifnum\beamer@tempcount>0
- \edef\inserttocsectionnumber{\the\beamer@tempcount}%
+ \ifnum\beamer@tempcount>0
+ \ifnum\beamer@tempcount>\beamer@toclastsection
+ \def\inserttocsectionnumber{}%
+ \else
+ \edef\inserttocsectionnumber{\the\beamer@tempcount}%
+ \fi
\else
\def\inserttocsectionnumber{}%
\fi%
diff --git a/macros/latex/contrib/beamer/base/themes/outer/beamerouterthemeinfolines.sty b/macros/latex/contrib/beamer/base/themes/outer/beamerouterthemeinfolines.sty
index e7bae70936..be53727f33 100644
--- a/macros/latex/contrib/beamer/base/themes/outer/beamerouterthemeinfolines.sty
+++ b/macros/latex/contrib/beamer/base/themes/outer/beamerouterthemeinfolines.sty
@@ -30,9 +30,14 @@
\begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
\usebeamerfont{title in head/foot}\insertshorttitle
\end{beamercolorbox}%
- \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%
- \usebeamerfont{date in head/foot}\insertshortdate{}\hspace*{2em}
- \usebeamercolor[fg]{page number in head/foot}\usebeamerfont{page number in head/foot}\usebeamertemplate{page number in head/foot}\hspace*{2ex}
+ \begin{beamercolorbox}[wd=.333333\paperwidth,ht=2.25ex,dp=1ex,leftskip=2ex,rightskip=2ex,sep=0pt]{date in head/foot}%
+ \hfill%
+ \usebeamerfont{date in head/foot}%
+ \insertshortdate{}%
+ \hfill%
+ \usebeamercolor[fg]{page number in head/foot}%
+ \usebeamerfont{page number in head/foot}%
+ \usebeamertemplate{page number in head/foot}%
\end{beamercolorbox}}%
\vskip0pt%
}
diff --git a/macros/latex/contrib/beamer/doc/beamerexample-conference-talk.pdf b/macros/latex/contrib/beamer/doc/beamerexample-conference-talk.pdf
index 9f9e5e7b62..8ee04e39a8 100644
--- a/macros/latex/contrib/beamer/doc/beamerexample-conference-talk.pdf
+++ b/macros/latex/contrib/beamer/doc/beamerexample-conference-talk.pdf
Binary files differ
diff --git a/macros/latex/contrib/beamer/doc/beamerexample-lecture-beamer-version.pdf b/macros/latex/contrib/beamer/doc/beamerexample-lecture-beamer-version.pdf
index 24fec63675..69bf38c2bf 100644
--- a/macros/latex/contrib/beamer/doc/beamerexample-lecture-beamer-version.pdf
+++ b/macros/latex/contrib/beamer/doc/beamerexample-lecture-beamer-version.pdf
Binary files differ
diff --git a/macros/latex/contrib/beamer/doc/beamerexample-lecture-print-version.pdf b/macros/latex/contrib/beamer/doc/beamerexample-lecture-print-version.pdf
index a4e92838cd..427b268ee3 100644
--- a/macros/latex/contrib/beamer/doc/beamerexample-lecture-print-version.pdf
+++ b/macros/latex/contrib/beamer/doc/beamerexample-lecture-print-version.pdf
Binary files differ
diff --git a/macros/latex/contrib/beamer/doc/beamerug-globalstructure.tex b/macros/latex/contrib/beamer/doc/beamerug-globalstructure.tex
index d24d9cf460..c76500d76f 100644
--- a/macros/latex/contrib/beamer/doc/beamerug-globalstructure.tex
+++ b/macros/latex/contrib/beamer/doc/beamerug-globalstructure.tex
@@ -548,6 +548,8 @@ sectionstyle=show/shaded,subsectionstyle=show/show/shaded
\item
\declare{|hideothersubsections|} causes the subsections of sections other than the current one to be hidden. This command is a shorthand for specifying the option |subsectionstyle=show/show/hide|.
\item
+ \declare{|lastsection=|\meta{section number}} similar to |firstsection|, this option specifies which section should be the last numbered section. This is useful if you have unnumbered sections at the end, like a summary or outlook. Section numbers are not shown by default. To show them, you must install a different table of contents templates.
+ \item
\declare{|part=|\meta{part number}} causes the table of contents of part \meta{part number} to be shown, instead of the table of contents of the current part (which is the default). This option can be combined with the other options, although combining it with the |current| option obviously makes no sense.
\item
\declare{|pausesections|} causes a |\pause| command to be issued before each section. This is useful if you wish to show the table of contents in an incremental way.
diff --git a/macros/latex/contrib/beamer/doc/beamerug-installation.tex b/macros/latex/contrib/beamer/doc/beamerug-installation.tex
index ef35ef53cc..bbee47a559 100644
--- a/macros/latex/contrib/beamer/doc/beamerug-installation.tex
+++ b/macros/latex/contrib/beamer/doc/beamerug-installation.tex
@@ -126,7 +126,7 @@ Note that if you have two versions installed, one in |texmf| and other in |texmf
To test your installation, copy the file |generic-ornate-15min-45min.en.tex| from the directory
\begin{verbatim}
-beamer/solutions/generic-talks
+beamer/doc/solutions/generic-talks
\end{verbatim}
to some place where you usually create presentations. Then run the command |pdflatex| several times on the file and check whether the resulting \pdf\ file looks correct. If so, you are all set.
diff --git a/macros/latex/contrib/beamer/doc/beamerug-localstructure.tex b/macros/latex/contrib/beamer/doc/beamerug-localstructure.tex
index b66371a703..3eb55d03e3 100644
--- a/macros/latex/contrib/beamer/doc/beamerug-localstructure.tex
+++ b/macros/latex/contrib/beamer/doc/beamerug-localstructure.tex
@@ -644,6 +644,13 @@ In the |article| version, the package |amsthm| sometimes clashes with the docume
Does not load |amssymb|. This option is mainly intended for users who are loading specialist font packages. Note that |\blacktriangleright| needs to be defined if |itemize| environments are in use.
\end{classoption}
+\begin{classoption}{leqno}
+ Place equation numbers on the left.
+\end{classoption}
+
+\begin{classoption}{fleqn}
+ Position equations at a fixed indent from the left margin rather than centered in the text column.
+\end{classoption}
\begin{element}{theorems}\semiyes\no\no
This template is a parent of |theorem begin| and |theorem end|, see the first for a detailed discussion of how the theorem templates are set.
diff --git a/macros/latex/contrib/beamer/doc/beamerug-solutions.tex b/macros/latex/contrib/beamer/doc/beamerug-solutions.tex
index b7ea537d99..50bdc1a554 100644
--- a/macros/latex/contrib/beamer/doc/beamerug-solutions.tex
+++ b/macros/latex/contrib/beamer/doc/beamerug-solutions.tex
@@ -12,11 +12,11 @@
\section{Solution Templates}
\label{section-solutions}
-In the subdirectories of the directory |beamer/solutions| you will find \emph{solution templates} in different languages. A solution template is a \TeX-text that ``solves'' a specific problem. Such a problem might be ``I need to create a 20 minute talk for a conference'' or ``I want to create a slide that introduces the next speaker'' or ``I want to create a table that is uncovered piecewise.'' For such a problem, a solution template consists of a mixture of a template and an example that can be used to solve this particular problem. Just copy the solution template file (or parts of it) and freely adjust them to your needs.
+In the subdirectories of the directory |beamer/doc/solutions| you will find \emph{solution templates} in different languages. A solution template is a \TeX-text that ``solves'' a specific problem. Such a problem might be ``I need to create a 20 minute talk for a conference'' or ``I want to create a slide that introduces the next speaker'' or ``I want to create a table that is uncovered piecewise.'' For such a problem, a solution template consists of a mixture of a template and an example that can be used to solve this particular problem. Just copy the solution template file (or parts of it) and freely adjust them to your needs.
The collecting of \beamer\ solution templates has only begun and currently there are only very few of them. We hope that in the future more solutions will become available and we would like to encourage users of the \beamer\ class to send us solutions they develop. We would also like to encourage users to help in translating solutions to languages other than English and German. If you have written a solution or a translation, please feel free to send it to us (make sure however, that it contains about the same amount of explanations and justifications as do the existing solutions).
-The following list of solution templates is sorted by the length of the talks for which they provide a template. As always, the solutions can be found in the directory |beamer/solutions|.
+The following list of solution templates is sorted by the length of the talks for which they provide a template. As always, the solutions can be found in the directory |beamer/doc/solutions|.
\begin{solution}{short-talks/speaker\_introduction-ornate-2min}
\begin{itemize}
diff --git a/macros/latex/contrib/beamer/doc/beamerug-tutorial.tex b/macros/latex/contrib/beamer/doc/beamerug-tutorial.tex
index 9dad541ac3..5f2dccc9b5 100644
--- a/macros/latex/contrib/beamer/doc/beamerug-tutorial.tex
+++ b/macros/latex/contrib/beamer/doc/beamerug-tutorial.tex
@@ -24,7 +24,7 @@ We wish to help Prof.\ Euclid of the University of Alexandria to create a presen
The first thing Euclid should do is to look for a solution template for his presentation. Having a look at Section~\ref{section-solutions}, he finds that the file
\begin{verbatim}
-beamer/solutions/conference-talks/conference-ornate-20min.en.tex
+beamer/doc/solutions/conference-talks/conference-ornate-20min.en.tex
\end{verbatim}
might be appropriate. He creates a subdirectory |presentation| in the directory that contains the actual paper and copies the solution template to this subdirectory, renaming to |main.tex|.
diff --git a/macros/latex/contrib/beamer/doc/beamerug-workflow.tex b/macros/latex/contrib/beamer/doc/beamerug-workflow.tex
index af9070bea6..d50dea12e4 100644
--- a/macros/latex/contrib/beamer/doc/beamerug-workflow.tex
+++ b/macros/latex/contrib/beamer/doc/beamerug-workflow.tex
@@ -20,7 +20,7 @@ This section presents a possible workflow for creating a \beamer\ presentation a
\beamernote
It is advisable that you create a folder for each presentation. Even though your presentation will usually reside in a single file, \TeX\ produces so many extra files that things can easily get very confusing otherwise. The folder's name should ideally start with the date of your talk in ISO format (like 2003-12-25 for a Christmas talk), followed by some reminder text of what the talk is all about. Putting the date at the front in this format causes your presentation folders to be listed nicely when you have several of them residing in one directory. If you use an extra directory for each presentation, you can call your main file |main.tex|.
-To create an initial |main.tex| file for your talk, copy an existing file from the |beamer/solutions| directory and adapt it to your needs. A list of possible \beamer\ solutions that contain templates for presentation \TeX-files can be found below.
+To create an initial |main.tex| file for your talk, copy an existing file from the |beamer/doc/solutions| directory and adapt it to your needs. A list of possible \beamer\ solutions that contain templates for presentation \TeX-files can be found below.
If you wish your talk to reside in the same file as some different, non-presentation article version of your text, it is advisable to setup a more elaborate file scheme. See Section~\ref{section-article-version-workflow} for details.
diff --git a/macros/latex/contrib/beamer/doc/beameruserguide.pdf b/macros/latex/contrib/beamer/doc/beameruserguide.pdf
index 838932df6e..d9e213862f 100644
--- a/macros/latex/contrib/beamer/doc/beameruserguide.pdf
+++ b/macros/latex/contrib/beamer/doc/beameruserguide.pdf
Binary files differ
diff --git a/macros/latex/contrib/beamer/doc/beameruserguide.tex b/macros/latex/contrib/beamer/doc/beameruserguide.tex
index fd0f7bcadf..b440db32f5 100644
--- a/macros/latex/contrib/beamer/doc/beameruserguide.tex
+++ b/macros/latex/contrib/beamer/doc/beameruserguide.tex
@@ -13,7 +13,7 @@
\documentclass{ltxdoc}
-\def\beamerugversion{3.62}
+\def\beamerugversion{3.63}
\def\beamerugpgfversion{1.00}
\def\beamerugxcolorversion{2.00}
@@ -64,8 +64,7 @@ User Guide for version \beamerugversion.}
\item<1-| alert@1> Suppose $p$ were the largest prime number.
\item<2-> Let $q$ be the product of the first $p$ numbers.
\item<3-> Then $q+1$ is not divisible by any of them.
- \item<1-> But $q + 1$ is greater than $1$, thus divisible by some prime
- number not in the first $p$ numbers.\qedhere
+ \item<1-> Thus $q+1$ is also prime and greater than $p$.\qedhere
\end{enumerate}
\end{proof}
\end{frame}