witharrows (17aug17)

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+The LaTeX extension witharrows is distributed under the LPPL 1.3 license.
+
+This extension has been written by F. Pantigny (fpantigny at wanadoo point fr).
+
+This extension is entirely contained in a single file : witharrows.sty
+
+The extension witharrows gives a LaTeX environment {WithArrows} which is similar
+to the environment {aligned} of amsmath but which gives the possibility to draw arrows 
+on the right side of the equations for explanations.
\ No newline at end of file
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+% This document is compatible with the following workflows :
+%    pdflatex,
+%    xelatex (with xdvipdfmx),
+%    latex --> dvips --> ps2pdf (or Adobe Distiller).
+\documentclass{article}
+
+\usepackage{ifxetex}
+
+\ifxetex
+  \usepackage{xltxtra}
+\else
+  \usepackage{expl3}
+  \usepackage{xparse}
+  \usepackage[T1]{fontenc}
+\fi
+
+\usepackage{xspace}
+
+
+\usepackage[dvipsnames]{xcolor}
+
+\usepackage{lmodern}
+
+\ifxetex
+   \usepackage[xetex]{geometry}
+\else
+   \usepackage{geometry}
+\fi
+
+\geometry{papersize={16.5cm,24cm},total={12.5cm,21cm},headheight=13.2pt,includeheadfoot,twoside,centering}
+
+
+\usepackage{tikz}
+\usetikzlibrary{calc}
+\usetikzlibrary{arrows.meta}
+\usetikzlibrary{bending}
+\usetikzlibrary{decorations.pathreplacing}
+
+
+\usepackage{footnote}
+
+\usepackage{witharrows}
+
+\usepackage{fancyvrb}
+\usepackage{mathtools}
+
+
+% this following lines only for this document
+
+\usepackage{amsfonts}
+
+\def\interitem{\vskip 7mm plus 2 mm minus 3mm}          
+\def\emphase#1{{\color{RoyalPurple}#1}}
+\fvset{commandchars=\~\#\@,formatcom={\color{gray}}}
+\DeclareMathOperator{\re}{Re}
+\let\group\begingroup
+
+
+\begin{document}
+
+
+\parindent0pt
+
+\title{Extension witharrows, v. 1.0}
+
+\author{Fran\c cois Pantigny}
+
+\date{2017/08/17}
+
+\maketitle
+
+
+
+\begin{abstract}
+The LaTeX package \verb|witharrows| gives an environment \verb|{WithArrows}| which is similar to environment
+\verb|{aligned}| of \verb|amsmath| (and \verb|mathtools|) but gives the possibility to draw arrows on the right
+side of the alignment. These arrows are usually used to give explanations concerning the mathematical calculus presented.
+\end{abstract}
+
+
+\vskip1cm
+
+This package uses Tikz to draw the arrows. It can be used with \verb|xelatex|, \verb|pdflatex| but also by the
+classical workflow \verb|latex|-\verb|dvips|-\verb|ps2pdf| (or Distiller). Two compilations may be necessary.
+
+\medskip
+This package gives an environment \verb|{WithArrows}| to construct alignments of equations with
+arrows for the explanations on the right side :
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 ~emphase#\Arrow{we expand}@ \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow{we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+
+
+\medskip
+The arrow has been drawn with the command \verb|\Arrow| on the ligne from which it starts. The command \verb|\Arrow|
+can be used anywhere on the line but the best way is to put it at the end.
+
+
+\medskip
+The ends of the lines of the tabular can be coded by \verb|\\| but also by \verb|\cr| (like in the command
+\verb|\halign| of TeX). In fact, in an environment \verb|{WithArrows}|, the command \verb|\\| has no option for the
+spacing between rows like in many environments of LaTeX.
+
+
+
+
+\section{Options for the shape of the arrows}
+
+The commande \verb|\Arrow| has several options. These options can be put between square brackets, before, or after
+the mandatory argument.
+
+The option \verb|jump| gives the number of lines the arrow must jump (the default value is, of course,~$1$)
+\footnote{It's not possible to give a non-positive value to \texttt{jump}. See below the way to draw an arrow which
+goes backwards.}
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 \Arrow~emphase#[jump=2]@{we expand} \\
+  & = (a+b)^2 + 2(a+b) +1 \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 \\
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 \Arrow[jump=2]{we expand} \\
+  & = (a+b)^2 + 2(a+b) +1 \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 \\
+\end{WithArrows}$
+
+
+
+
+\interitem
+It's possible to put several arrows which start from the same line.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 ~emphase#\Arrow{}\Arrow{}[jump=2]@ \\
+  & = (a+b)^2 + 2(a+b) +1 \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 \Arrow{}\Arrow{}[jump=2] \\
+  & = (a+b)^2 + 2(a+b) +1 \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 
+\end{WithArrows}$
+
+
+\interitem
+The option \verb|xoffset| shift the arrows to the right (we usually don't want the arrows to be stucked
+on the text). The default value of \verb|xoffset| is $3$~mm.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 
+\Arrow[~emphase#xoffset=1cm@]{with \texttt{xoffset=1cm}} \\
+  & = (a+b)^2 + 2(a+b) +1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 
+\Arrow[xoffset=1cm]{with \texttt{xoffset=1cm}} \\
+  & = (a+b)^2 + 2(a+b) +1 \\
+\end{WithArrows}$
+
+
+
+
+\interitem 
+The arrows are drawn with Tikz. That's why the command \verb|\Arrow| has an option
+\verb|tikz| which can be used to give to the arrow (in fact, the command \verb|\path| of Tikz) the options
+proposed by Tikz for such an arrow. The following example gives an blue thick arrow.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[~emphase#tikz={blue,thick}@]{we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[tikz={blue,thick}]{we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+
+
+\interitem
+It's also possible to change the arrowheadss. For example, we can draw an arrow which goes backwards 
+with the Tikz option~\verb|<-|.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow~emphase#[tikz=<-]@{we factorize} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[tikz=<-]{we factorize} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+
+
+\interitem
+It's also possible to suppress both tips of the arrow with the Tikz option \verb|-|.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[~emphase#tikz=-@]{very classical} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[tikz=-]{very classical} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+
+
+\interitem
+In order to have straight arrows instead of curved ones, we must use the Tikz option ``\verb|bend left = 0|''.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow~emphase#[tikz={bend left=0}]@{we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[tikz={bend left=0}]{we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+
+
+\interitem
+One of the most useful options is ``\verb|text width|'' to control the with of the text associated to the arrow. 
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 
+\Arrow[jump=2,~emphase#tikz={text width=5.3cm}@]{We have done...} \\
+  & = (a+b)^2 + 2(a+b) +1 \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = \bigl((a+b)+1\bigr)^2 
+\Arrow[jump=2,tikz={text width=5.3cm}]{We have done a two-stages expansion but it would have been clever
+to expand with the multinomial theorem.} \\
+  & = (a+b)^2 + 2(a+b) +1 \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 
+\end{WithArrows}$
+
+
+\interitem
+If we want to change the font of the text associated to the arrow, we can, of course, put a command like
+\verb|\bfseries|, \verb|\large| or \verb|\sffamily| at the beginning of the text. But, by default, the texts
+are composed with a combination of \verb|\small| and \verb|\itshape|. When adding \verb|\bfseries| at the beginning
+of the text, we won't suppress the \verb|\small| and the \verb|\itshape| and we will consequently have a text in a
+bold, italic and small font.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow{~emphase#\bfseries@ we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow{\bfseries we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+
+\interitem 
+If we put commands \verb|\\| in the text to force newlines, a command of font placed in the beginning of the
+text will have effect only until the first command \verb|\\| (like in a environment \verb|{tabular}|). That's why
+Tikz gives a option \verb|font| to modify the font of the whole text. Nevertheless, if we use the option
+\verb|tikz={font={\bfseries}}|, the default specification of \verb|\small| and \verb|\itshape| will be overwritten.
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[~emphase#tikz={font={\bfseries}}@]{we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow[tikz={font={\bfseries}}]{we expand} \\
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+\medskip
+If we want exactly the same result as previously, we have to give to the option \verb|font| the value 
+\verb|{\itshape\small\bfseries}|.
+
+
+
+
+\interitem 
+Almost all the options can be given directly to the environment \verb|WithArrows| (between square brackets).
+In this case, they apply to all the arrows of the environment.
+%
+\begin{Verbatim}
+$\begin{WithArrows}~emphase#[tikz=blue]@
+A & = \bigl((a+b)+1\bigr)^2 \Arrow{First expansion.} \\
+  & = (a+b)^2 + 2(a+b) +1 \Arrow{Second expansion.} \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}[tikz=blue]
+A & = \bigl((a+b)+1\bigr)^2 \Arrow{First expansion.} \\
+  & = (a+b)^2 + 2(a+b) +1 \Arrow{Second expansion.} \\
+  & = a^2 + 2ab + b^2 + 2a + 2b +1 
+\end{WithArrows}$
+
+
+\interitem
+The environment \verb|{WithArrows}| has an option \verb|displaystyle|. With this option, all the elements are
+composed in \verb|\displaystyle| (like in an environment \verb|{aligned}| of \verb|amsmath|). 
+
+\medskip
+Without the option \verb|displaystyle| :
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+\int_0^1 (x+1)^2 dx 
+& = \int_0^1 (x^2+2x+1) dx
+\Arrow{linearity of integration}     \\
+& = \int_0^1 x^2 dx + 2 \int_0^1 x dx + \int_0^1 dx \\
+& = \frac13 + 2\frac12 + 1 \\
+& = \frac73
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+\int_0^1 (x+1)^2 dx 
+& = \int_0^1 (x^2+2x+1) dx
+\Arrow{linearity of integration}     \\
+& = \int_0^1 x^2 dx + 2 \int_0^1 x dx + \int_0^1 dx \\
+& = \frac13 + 2\frac12 + 1 \\
+& = \frac73
+\end{WithArrows}$
+
+
+\medskip
+The same example with the option \verb|displaystyle| :\par\nobreak
+
+$\begin{WithArrows}[displaystyle]
+\int_0^1 (x+1)^2 dx 
+& = \int_0^1 (x^2+2x+1) dx
+\Arrow{linearity of integration}     \\
+& = \int_0^1 x^2 dx + 2 \int_0^1 x dx + \int_0^1 dx \\
+& = \frac13 + 2\frac12 + 1 \\
+& = \frac73
+\end{WithArrows}$
+
+
+\interitem 
+Almost all the options can also be set at the document level with the command \verb|\WithArrowsOptions|. 
+In this case, the scope of the declarations is the current TeX group (these declarations are ``semi-global'').
+For example, if we want all the environments \verb|{WithArrows}| composed in \verb|\displaystyle| with blue
+arrows, we can write \verb|\WithArrowsOptions{displaystyle,tikz=blue}|.\footnote{It's also possible to give the
+  options directly when loading the package, \emph{i.e.} with the command \texttt{\string\usepackage} in the preamble.}
+%
+\begin{Verbatim}
+~emphase#\WithArrowsOptions{displaystyle,tikz=blue}@
+$\begin{WithArrows}
+\sum_{i=1}^n (x_i+1)^2 
+& = \sum_{i=1}^n (x_i^2+2x_i+1) \Arrow{by linearity}\\
+& = \sum_{i=1}^n x_i^2 + 2\sum_{i=1}^nx_i+ n
+\end{WithArrows}$
+\end{Verbatim}
+
+\begin{group}
+\WithArrowsOptions{displaystyle,tikz=blue}
+$\begin{WithArrows}
+\sum_{i=1}^n (x_i+1)^2 
+& = \sum_{i=1}^n (x_i^2+2x_i+1) \Arrow{by linearity}\\
+& = \sum_{i=1}^n x_i^2 + 2\sum_{i=1}^nx_i+ n
+\end{WithArrows}$
+\end{group}
+
+
+\interitem
+The command \verb|\Arrow| is recognized only in the environments \verb|{WithArrows}|. If we have a command 
+\verb|\Arrow| previously defined, it's possible to go on using it outside the environments \verb|{WithArrows}|.
+
+However, a previouly defined command \verb|\Arrow| may still be useful in a environment \verb|{WithArrows}|. If we
+want to use it in such an environment, it's possible to change the name of the command \verb|\Arrow| of the package
+\verb|witharrows| : there is an option \verb|CommandName| for this purpose. The new name of the command must be
+given to the option \emph{without} the leading backslash.
+%
+\begin{Verbatim}
+\def\Arrow{\longmapsto}
+$\begin{WithArrows}~emphase#[CommandName=Explanation]@
+f & = \bigl(x \Arrow (x+1)^2\bigr)
+~emphase#\Explanation{we work directly on fonctions}@\\
+& = \bigl(x \Arrow x^2+2x+1\bigr)
+\end{WithArrows}$
+\end{Verbatim}
+%
+\begin{group}
+\def\Arrow{\longmapsto}
+$\begin{WithArrows}[CommandName=Explanation]
+f & = \bigl(x \Arrow (x+1)^2\bigr)
+\Explanation{we work directly on fonctions}\\
+& = \bigl(x \Arrow x^2+2x+1\bigr)
+\end{WithArrows}$
+\end{group}
+
+
+\interitem 
+It's possible to use directly the nodes created by \verb|{WithArrows}| (see below) with explicit Tikz instructions
+(in order, for example, to draw something that can't be drawn with the command \verb|\Arrow|). That's why a style
+for the tips of the arrows has be created : \verb|TipsOfWithArrows|. By using this style, we will have homogeneous
+tips for the arrows of the document.
+
+Therefore, if we want to  modify the tips of the arrows of \verb|{WithArrows}|, we have to modify the style
+\verb|TipsOfWithArrows|. 
+%
+\begin{Verbatim}
+\tikzset{TipsOfWithArrows/.style= { > = {Latex[scale=1.2,bend]}} }
+\end{Verbatim}
+
+
+
+
+\section{Precise positioning of the arrows}
+
+
+The environment \verb|{WithArrows}| defines, during the composition of the array, two series of nodes materialized in
+red in the following example.\footnote{The option \texttt{shownodes} can be used to materialize the nodes.}
+
+\smallskip
+$\begin{WithArrows}[displaystyle,shownodes]
+I
+& = \int_{\frac{\pi}4}^0 \ln\Bigl(1+\tan\left(\tfrac{\pi}4-u\right)\Bigr)(-d u) \\
+& = \int_0^{\frac{\pi}4} \ln\Bigl(1+\tan\left(\tfrac{\pi}4-u\right)\Bigr)d u \\
+& = \int_0^{\frac{\pi}4}\ln\left(1+\frac{1-\tan u}{1+\tan u}\right)\, d u \\
+& =\int_0^{\frac{\pi}4}\ln\left(\frac{1+\tan u+1-\tan u}{1+\tan u}\right) d u \\
+& =\int_0^{\frac{\pi}4} \ln\left(\frac2{1+\tan u}\right)\, d u\\
+& =\int_0^{\frac{\pi}4}\bigl(\ln2-\ln(1+\tan u)\bigr)\, d u \\
+& =\frac{\pi}4\ln2-\int_0^{\frac{\pi}4}\ln(1+\tan u)\, d u  \\
+& =\frac{\pi}4\ln2-I 
+\end{WithArrows}$
+
+\bigskip
+The nodes of the left are at the end of each line of text. These nodes will be called \emph{left nodes}.
+The nodes of the right side are aligned vertically on the right side of the array. These nodes will be called
+\emph{right nodes}.\footnote{The names of the Tikz nodes created by \texttt{\{WithArrows\}} are \texttt{wa-}$n$\texttt{-l} and
+  \texttt{wa-}$n$\texttt{-r} where $n$~is the number of the line. It's possible to refer to these Tikz nodes after the
+  environment (one should use the options \texttt{remember picture} and \texttt{overlay} and also
+  \texttt{TipsOfWithArrows} and \texttt{->} in order to have the same arrowheads).}
+
+By default, the arrows use the right nodes. We will say that they are in \verb|rr| mode ($r$ for \emph{right}).
+These arrows are \verb|vertical| (we will say that an arrow is \emph{vertical} when its two ends have the
+same abscissa).
+
+
+\smallskip
+However, it's possible to use the left nodes, or a combination of left and right nodes, with one of the options
+\verb|lr|, \verb|rl| and \verb|ll| ($l$ for \emph{left}). Those arrows are, usually, not vertical.
+
+
+Therefore
+$\begin{WithArrows}[displaystyle]
+I
+& = \int_{\frac{\pi}4}^0 \ln\Bigl(1+\tan\left(\tfrac{\pi}4-u\right)\Bigr)(-d u) 
+\Arrow[lr]{This arrow uses the \texttt{lr} option.}\\
+& = \int_0^{\frac{\pi}4} \ln\Bigl(1+\tan\left(\tfrac{\pi}4-u\right)\Bigr)d u \\
+& = \int_0^{\frac{\pi}4}\ln\left(1+\frac{1-\tan u}{1+\tan u}\right)\, d u \\
+& =\int_0^{\frac{\pi}4}\ln\left(\frac{1+\tan u+1-\tan u}{1+\tan u}\right) d u 
+\Arrow[ll,jump=2,tikz={text width = 5cm}]{This arrow uses a \texttt{ll} option and a \texttt{jump} equal to $2$}\\
+& =\int_0^{\frac{\pi}4} \ln\left(\frac2{1+\tan u}\right)\, d u\\
+& =\int_0^{\frac{\pi}4}\bigl(\ln2-\ln(1+\tan u)\bigr)\, d u \\
+& =\frac{\pi}4\ln2-\int_0^{\frac{\pi}4}\ln(1+\tan u)\, d u  \\
+& =\frac{\pi}4\ln2-I 
+\end{WithArrows}$
+
+
+
+\interitem
+There is also an option called \texttt{i} ($i$ for \emph{intermediate}). With this option, the arrow is vertical
+and at the leftmost position.
+
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+(a+b)(a+ib)(a-b)(a-ib) 
+& = (a+b)(a-b)\cdot(a+ib)(a-ib) \\
+& = (a^2-b^2)(a^2+b^2) \Arrow~emphase#[i]@{because $(x-y)(x+y)=x^2-y^2$}\\
+& = a^4-b^4 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+(a+b)(a+ib)(a-b)(a-ib) 
+& = (a+b)(a-b)\cdot(a+ib)(a-ib) \\
+& = (a^2-b^2)(a^2+b^2) \Arrow[i]{because $(x-y)(x+y)=x^2-y^2$}\\
+& = a^4-b^4 
+\end{WithArrows}$
+
+
+\interitem 
+The environment \verb|{WithArrows}| gives also a \verb|group| option. With this option, \emph{all} the
+arrows of the environment are grouped on a same vertical line and at a leftmost position.
+%
+\begin{Verbatim}[formatcom=\small\color{gray}]
+$\begin{WithArrows}~emphase#[displaystyle,group]@
+2xy'-3y=\sqrt x
+& \Longleftrightarrow 2x(K'y_0+Ky_0')-3Ky_0 = \sqrt x \\
+& \Longleftrightarrow 2xK'y_0 + K(2xy_0'-3y_0) = \sqrt x \\
+& \Longleftrightarrow 2x K'y_0 = \sqrt x \Arrow{...}\\
+...
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}[displaystyle,group]
+2xy'-3y=\sqrt x
+& \Longleftrightarrow 2x(K'y_0+Ky_0')-3Ky_0 = \sqrt x \\
+& \Longleftrightarrow 2xK'y_0 + K(2xy_0'-3y_0) = \sqrt x \\
+& \Longleftrightarrow 2x K'y_0 = \sqrt x \Arrow{We remplace $y_0$ by its value.}\\
+& \Longleftrightarrow 2xK'x^{\frac32} = x^{\frac12} \Arrow{simplification of the $x$}\\
+& \Longleftrightarrow K' = \tfrac1{2x²} \Arrow{antiderivation}\\
+& \Longleftrightarrow K = -\tfrac1{2x} 
+\end{WithArrows}$
+
+
+\bigskip
+If desired, the option \verb|group| can be given to the command \verb|WithArrowsOptions| so that it will become the
+default value.
+
+
+\section{Comparison with the environment \{aligned\}}
+
+
+
+\verb|{WithArrows}| bears similarities with the environment \verb|{aligned}| of the extension
+\verb|amsmath|. These are only similarities because \verb|{WithArrows}| has not been written upon the environment
+\verb|{aligned}|. 
+
+In paticular, the command \verb|\\| of the end of line has no option to change the row spacings. In fact, this
+command \verb|\\| can be replaced, in a environment \verb|{WithArrows}| by a \verb|\cr| of TeX. That's what we will 
+do in the balance of this document.
+
+
+
+
+\interitem
+If desired, it's possible to change the spacing between two given lines by putting a command
+\verb|\vspace| in a \verb|\noalign| (which is a low level command of TeX to insert extraordinary elements between
+the lines of an array).
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow{we expand} \cr
+~emphase#\noalign{\vspace{3mm}}@
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow{we expand} \cr
+\noalign{\vspace{3mm}}
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+
+\interitem 
+In the  environments of \verb|amsmath|, the spacing between lines is fixed by a parameter called
+\verb|\jot| and that's also the case for the environment \verb|{WithArrows}|. An option \verb|jot| has been given
+to the environment \verb|{WithArrows}| in order to change the value of this parameter \verb|\jot| for an given environment.
+%
+\begin{Verbatim}
+$\begin{WithArrows}[displaystyle,~emphase#jot=2mm@]
+F & = \frac12G     \Arrow{we expand}\cr
+  & = H + \frac12K \Arrow{we go on}\cr
+  & = K 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}[displaystyle,jot=2mm]
+F & = \frac12G     \Arrow{we expand}\cr
+  & = H + \frac12K \Arrow{we go on}\cr
+  & = K 
+\end{WithArrows}$
+
+
+\interitem
+Like the environment \verb|{aligned}|, \verb|{WithArrows}| has an option of placement which can assume the values
+\verb|t|, \verb|c| or \verb|b|. However, the default value is not \verb|c| but \verb|t|. If desired, it's possible
+to have the \verb|c| value as the default with the command \verb|WithArrowsOptions{c}| at the beginning of the document.
+%
+\begin{Verbatim}
+~emphase#Et donc\enskip@
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow{we expand} \cr
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+\end{Verbatim}
+
+Et donc\enskip 
+$\begin{WithArrows}
+A & = (a+1)^2 \Arrow{we expand} \cr
+  & = a^2 + 2a + 1 
+\end{WithArrows}$
+
+\bigskip
+The value \verb|c| may be useful, for example, if we want to add curly braces :
+
+\smallskip
+\begin{Verbatim}
+On pose\enskip $~emphase#\left\{@
+\begin{WithArrows}~emphase#[c]@
+f(x) & = 3x^3+2x^2-x+4 
+\Arrow[tikz=-]{both are polynoms}\cr
+g(x) & = 5x^2-5x+6
+\end{WithArrows}
+~emphase#\right.@$
+\end{Verbatim}
+
+
+
+
+On pose\enskip $\left\{
+\begin{WithArrows}[c]
+f(x) & = 3x^3+2x^2-x+4 
+\Arrow[tikz=-]{both are polynoms}\cr
+g(x) & = 5x^2-5x+6
+\end{WithArrows}
+\right.$
+
+
+\interitem
+Unlike \verb|{aligned}|, the environment \verb|{WithArrows}| uses \verb|\textstyle| by default.
+
+Once again, it's possible to change this behaviour with \verb|WithArrowsOptions| :
+
+\quad \verb|WithArrowsOptions{displaystyle}|.
+
+
+\smallskip
+The following example is composed with \verb|{aligned}| :\par\nobreak
+
+\smallskip
+$\left\{
+\begin{aligned}
+\sum_{i=1}^n (x_i+1)^2  
+& = \sum_{i=1}^n (x_i^2 + 2x_i+1) \\
+& = \sum_{i=1}^n x_i^2 + 2 \sum_{i=1}^nx_i + n 
+\end{aligned}
+\right.$%
+
+
+\medskip
+The following is composed with \verb|{WithArrows}[c,displaystyle]|. The results are stricly identical.\par\nobreak
+
+\smallskip
+$\left\{
+\begin{WithArrows}[c,displaystyle]
+\sum_{i=1}^n (x_i+1)^2 
+& = \sum_{i=1}^n (x_i^2 + 2x_i+1) \\
+& = \sum_{i=1}^n x_i^2 + 2 \sum_{i=1}^nx_i + n 
+\end{WithArrows}
+\right.$%
+
+
+
+
+\section{Examples}
+
+It's possible to use the environment \verb|{WithArrows}| with making use of the left column only, or the right
+column only. 
+%
+\begin{Verbatim}
+$\begin{WithArrows}
+&f(x) \ge g(x) \Arrow{by squaring both sides} \cr
+& f(x)^2 \ge g(x)^2 \Arrow{by moving to left side} \cr
+& f(x)^2 - g(x)^2 \ge 0 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}
+&f(x) \ge g(x) \Arrow{by squaring both sides} \cr
+& f(x)^2 \ge g(x)^2 \Arrow{by moving to left side} \cr
+& f(x)^2 - g(x)^2 \ge 0 
+\end{WithArrows}$
+
+
+\interitem
+Here is an example with a loop flow.
+%
+\begin{Verbatim}[formatcom=\small\color{gray}]
+$\begin{WithArrows}[tikz={font={\tiny}}]
+a.\;& f \text{ est continuous on } E 
+\Arrow{(1)}\Arrow[tikz=<-,jump=4,xoffset=1cm]{(5)}\cr
+b.\;& f \text{ est continuous in } 0 
+\Arrow{(2)}\cr
+c.\;& f \text{ is bounded on the unit sphere} 
+\Arrow{(3)}\cr
+d.\;& \exists K > 0\quad \forall x \in E\quad \|f(x)\| \le K \|x\| 
+\Arrow{(4)}\cr
+e.\;& f \text{ is lipschitzian} 
+\end{WithArrows}$
+\end{Verbatim}
+
+$\begin{WithArrows}[tikz={font={\tiny}}]
+a.\;& f \text{ est continuous on } E 
+\Arrow{(1)}\Arrow[tikz=<-,jump=4,xoffset=1cm]{(5)}\cr
+b.\;& f \text{ est continuous in } 0 
+\Arrow{(2)}\cr
+c.\;& f \text{ is bounded on the unit sphere} 
+\Arrow{(3)}\cr
+d.\;& \exists K > 0\quad \forall x \in E\quad \|f(x)\| \le K \|x\| 
+\Arrow{(4)}\cr
+e.\;& f \text{ is lipschitzian} 
+\end{WithArrows}$
+
+\interitem
+The option \verb|font| of Tikz contains in fact a list of tokens which will be placed at the beginning of the text.
+
+These tokens can be true commands for a changement of font (like \verb|\bfseries| or \verb|\sffamily|) but can also
+be, in fact, any TeX command.
+
+In the following example, the argument of \verb|font| is the token list \verb|\tiny\counter| where
+\verb|\counter| is a command which increment a counter previously defined and display its new value. Thus, the
+arrows are automatically numbered.
+%
+\begin{Verbatim}[formatcom=\small\color{gray}]
+\newcounter{MyCounter}
+\newcommand{\counter}{\stepcounter{MyCounter}\theMyCounter.}
+$\begin{WithArrows}[tikz={~emphase#font={\tiny\counter}@}]
+A(x) 
+& = B(x) \Arrow{} \cr
+& = C(x) \Arrow{} \cr
+& = C(x) \Arrow{} \cr
+& = E(x) \Arrow{} \cr
+& = F(x) \Arrow{} \cr
+& = G(x)   
+\end{WithArrows}$
+\end{Verbatim}
+
+\begin{group}
+\newcounter{MyCounter}
+\newcommand{\counter}{\stepcounter{MyCounter}\theMyCounter.}
+$\begin{WithArrows}[tikz={font={\tiny\counter}}]
+A(x) 
+& = B(x) \Arrow{} \cr
+& = C(x) \Arrow{} \cr
+& = C(x) \Arrow{} \cr
+& = E(x) \Arrow{} \cr
+& = F(x) \Arrow{} \cr
+& = G(x)   
+\end{WithArrows}$
+\end{group}
+
+
+
+\end{document}
+
+
+
+
+