\documentclass[rep10,leqno]{report}
\voffset = -1in
\evensidemargin 0.1in
\oddsidemargin 0.1in
\textheight 9in
\textwidth 6in
\begin{document}
\normalsize
\baselineskip=12pt
\noindent
Input = 
$
\\A\,
 Pinv[A]\,
 A == A\\
Pinv[A]\,
 A\,
 Pinv[A] == Pinv[A]\\
A{}^{ T }\,
 \left(Pinv[A]\right){}^{ T } == Pinv[A]\,
 A\\
\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } == A\,
 Pinv[A]\\
\left(Pinv[A]\right){}^{ T } == Pinv[A{}^{ T }]\\
A{}^{ T }\,
 A\,
 Pinv[A] == A{}^{ T }\\
A\,
 Pinv[A]\,
 Pinv[A{}^{ T }] == Pinv[A{}^{ T }]\\
B\,
 Pinv[B]\,
 B == B\\
Pinv[B]\,
 B\,
 Pinv[B] == Pinv[B]\\
B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T } == Pinv[B]\,
 B\\
\left(Pinv[B]\right){}^{ T }\,
 B{}^{ T } == B\,
 Pinv[B]\\
\left(Pinv[B]\right){}^{ T } == Pinv[B{}^{ T }]\\
B{}^{ T }\,
 B\,
 Pinv[B] == B{}^{ T }\\
B\,
 Pinv[B]\,
 Pinv[B{}^{ T }] == Pinv[B{}^{ T }]\\
(A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A])\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 (A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]) == A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]\\
Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 (A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A])\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]] == Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\\
(\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T })\,
 \left(Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\right){}^{ T } == Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 (A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A])\\
\left(Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\right){}^{ T }\,
 (\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }) == (A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A])\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\\
\left(Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\right){}^{ T } == Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\\
(\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T })\,
 (A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A])\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]] == \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }\\
(A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A])\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }] == Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\\
$
File Name = APinvAMinusPinvBBAPinvA\\
NCMakeGB Iterations = 6\\
NCMakeGB on Digested Iterations = 0\\
SmallBasis Iterations = 0\\
SmallBasis on Knowns Iterations = 0\\
Deselect$\rightarrow \{\}$\\
UserSelect$\rightarrow \{\}$\\
UserUnknowns$\rightarrow \{\}$\\
NCShortFormulas$\rightarrow$-1\\
RR$\rightarrow $False\\
RRByCat$\rightarrow $False\\
SB$\rightarrow $False\\
SBByCat$\rightarrow $False\\
DegreeCap$\rightarrow $-1\\
DegreeSumCap$\rightarrow $-1\\
DegreeCapSB$\rightarrow $-1\\
DegreeSumCapSB$\rightarrow $-1\\
NCCV$\rightarrow $False\\
THE ORDER IS NOW THE FOLLOWING:\hfil\break
$
A<B<A{}^{ T }<B{}^{ T }<Pinv[A]<Pinv[B]<Pinv[A{}^{ T }]<Pinv[B{}^{ T }]<\left(Pinv[A]\right){}^{ T }<\left(Pinv[B]\right){}^{ T }<Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]<Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]<\left(Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\right){}^{ T }\\
$
\rule[2pt]{6in}{4pt}\hfil\break
\rule[2pt]{1.879in}{4pt}
\ YOUR SESSION HAS DIGESTED\ 
\rule[2pt]{1.879in}{4pt}\hfil\break
\rule[2pt]{1.923in}{4pt}
\ THE FOLLOWING RELATIONS\ 
\rule[2pt]{1.923in}{4pt}\hfil\break
\rule[2pt]{6in}{4pt}\hfil\break
THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR:\hfil\break
$\{\left(Pinv[A]\right){}^{ T },
$ $
\left(Pinv[B]\right){}^{ T },
$ $
\left(Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\right){}^{ T }\}$
\smallskip\\
The corresponding rules are the following:\smallskip\\
\begin{minipage}{6in}
$
\left(Pinv[A]\right){}^{ T }\rightarrow Pinv[A{}^{ T }]
$
\end{minipage}\medskip\\
\begin{minipage}{6in}
$
\left(Pinv[B]\right){}^{ T }\rightarrow Pinv[B{}^{ T }]
$
\end{minipage}\medskip\\
\begin{minipage}{6in}
$
\left(Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\right){}^{ T }\rightarrow Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]
$
\end{minipage}\medskip\\
\rule[3pt]{6in}{.7pt}\\
The expressions with unknown variables $\{\}$\\
and knowns $\{A,
$ $
B,
$ $
Pinv[A],
$ $
Pinv[B],
$ $
Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]],
$ $
Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }],
$ $
Pinv[A{}^{ T }],
$ $
Pinv[B{}^{ T }],
$ $
A{}^{ T },
$ $
B{}^{ T }\}$\smallskip\\
\begin{minipage}{6in}
$
B\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\rightarrow 0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A]\,
 A\rightarrow A{}^{ T }\,
 Pinv[A{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[B]\,
 B\rightarrow B{}^{ T }\,
 Pinv[B{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[B]\rightarrow 0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\rightarrow Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 B{}^{ T }\rightarrow 0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\rightarrow Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A{}^{ T }]\,
 A{}^{ T }\rightarrow A\,
 Pinv[A]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[B{}^{ T }]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\rightarrow 0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[B{}^{ T }]\,
 B{}^{ T }\rightarrow B\,
 Pinv[B]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A\,
 Pinv[A]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\rightarrow Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A\,
 Pinv[A]\,
 Pinv[A{}^{ T }]\rightarrow Pinv[A{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A\,
 A{}^{ T }\,
 Pinv[A{}^{ T }]\rightarrow A
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B\,
 Pinv[B]\,
 Pinv[B{}^{ T }]\rightarrow Pinv[B{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B\,
 B{}^{ T }\,
 Pinv[B{}^{ T }]\rightarrow B
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 A\rightarrow Pinv[A]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 A
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\rightarrow Pinv[A]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[A{}^{ T }]\rightarrow Pinv[A]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[A{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 A\,
 Pinv[A]\rightarrow A\,
 Pinv[A]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 A\,
 Pinv[A]\rightarrow Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 A\rightarrow Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 A
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[A{}^{ T }]\rightarrow Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[A{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 B{}^{ T }\,
 Pinv[B{}^{ T }]\rightarrow B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]] - Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]] + Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A{}^{ T }\,
 A\,
 Pinv[A]\rightarrow A{}^{ T }
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A{}^{ T }\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 A\rightarrow A{}^{ T }\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 A
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A{}^{ T }\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\rightarrow A{}^{ T }\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A{}^{ T }\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\,
 Pinv[A{}^{ T }]\rightarrow A{}^{ T }\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[A{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A{}^{ T }\,
 Pinv[A{}^{ T }]\,
 Pinv[A]\rightarrow Pinv[A]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B{}^{ T }\,
 B\,
 Pinv[B]\rightarrow B{}^{ T }
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[B]\rightarrow Pinv[B]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
Pinv[A]\,
 B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\rightarrow Pinv[A]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]] + Pinv[A]\,
 B{}^{ T }\,
 Pinv[B{}^{ T }] - Pinv[A]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
A{}^{ T }\,
 B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\rightarrow A{}^{ T }\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]] + A{}^{ T }\,
 B{}^{ T }\,
 Pinv[B{}^{ T }] - A{}^{ T }
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 A\rightarrow -1\,
 A + Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 A + B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 A
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\rightarrow Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }] - Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[A{}^{ T }]\rightarrow Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]]\,
 Pinv[A{}^{ T }] + B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 Pinv[A{}^{ T }] - Pinv[A{}^{ T }]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 A\,
 Pinv[A]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }]\rightarrow -1\,
 A\,
 Pinv[A] + A\,
 Pinv[A]\,
 Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }] + B{}^{ T }\,
 Pinv[B{}^{ T }]\,
 A\,
 Pinv[A]
$
\end{minipage}\\
\rule[2pt]{6in}{1pt}\hfil\break
\rule[2.5pt]{1.701in}{1pt}
\ USER CREATIONS APPEAR BELOW\ 
\rule[2.5pt]{1.701in}{1pt}\hfil\break
\rule[2pt]{6in}{1pt}\hfil\break
\rule[2pt]{6in}{4pt}\hfil\break
\rule[2pt]{1.45in}{4pt}
\ SOME RELATIONS WHICH APPEAR BELOW\ 
\rule[2pt]{1.45in}{4pt}\hfil\break
\rule[2pt]{2.18in}{4pt}
\ MAY BE UNDIGESTED\ 
\rule[2pt]{2.18in}{4pt}\hfil\break
\rule[2pt]{6in}{4pt}\hfil\break
THE FOLLOWING VARIABLES HAVE NOT BEEN SOLVED FOR:\hfil\break
$\{A,
$ $
B,
$ $
Pinv[A],
$ $
Pinv[B],
$ $
Pinv[A\,
 Pinv[A] - Pinv[B]\,
 B\,
 A\,
 Pinv[A]],
$ $
Pinv[\left(Pinv[A]\right){}^{ T }\,
 A{}^{ T } - \left(Pinv[A]\right){}^{ T }\,
 A{}^{ T }\,
 B{}^{ T }\,
 \left(Pinv[B]\right){}^{ T }],
$ $
Pinv[A{}^{ T }],
$ $
Pinv[B{}^{ T }],
$ $
A{}^{ T },
$ $
B{}^{ T }\}$
\smallskip\\
\vspace{10pt}

\noindent
The time for preliminaries was 0:00:00\\
The time for NCMakeGB 1 was 0:00:00\\
The time for Remove Redundant 1 was 0:00:00\\
The time for the main NCMakeGB was 0:00:04\\
The time for Remove Redundant 2 was 0:00:00\\
The time for reducing unknowns was 0:00:00\\
The time for clean up basis was 0:00:00\\
The time for SmallBasis was 0:00:00\\
The time for CreateCategories was 0:00:04\\
The time for NCCV was 0:00:00\\
The time for RegularOutput was 0:00:00\\
The time for everything so far was 0:00:08\\
\end{document}
\end