# LaTeX2HTML 2022 (Released January 1, 2022)
# Associate images original text with physical files.
$key = q/displaystylesqrt{{d_{max}^2+d_{min}^2}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 4.27ex; vertical-align: -1.40ex; " SRC="|."$dir".q|img4.png"
 ALT="$\displaystyle \sqrt{{d_{\max}^2+d_{\min}^2}}$">|; 
$key = q/lvert;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.38ex; vertical-align: -0.70ex; " SRC="|."$dir".q|img1.png"
 ALT="$\lvert$">|; 
$key = q/nolimits;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 0.21ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img3.png"
 ALT="$\nolimits$">|; 
$key = q/rvert^{2}_{};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 STYLE="height: 2.59ex; vertical-align: -0.70ex; " SRC="|."$dir".q|img2.png"
 ALT="$\rvert^{2}_{}$">|; 
$key = q/{proof}%latex2htmlidmarker113[ProofofTheoremref{pigspan}]Andanothertest.{proof};AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="height: 1.75ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img6.png"
 ALT="\begin{proof}
% latex2html id marker 113
[Proof of Theorem \ref{pigspan}]
And another test.
\end{proof}">|; 
$key = q/{proof}Hereisatestoftheproofenvironment.{proof};AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="height: 2.17ex; vertical-align: -0.57ex; " SRC="|."$dir".q|img5.png"
 ALT="\begin{proof}
Here is a test of the proof environment.
\end{proof}">|; 
$key = q/{proof}[Proof(necessity)]Andanother.{proof};AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="height: 1.75ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img7.png"
 ALT="\begin{proof}[Proof (necessity)]
And another.
\end{proof}">|; 
$key = q/{proof}[Proof(sufficiency)]Andanother.{proof};AAT/;
$cached_env_img{$key} = q|<IMG
  STYLE="height: 1.75ex; vertical-align: -0.12ex; " SRC="|."$dir".q|img8.png"
 ALT="\begin{proof}[Proof (sufficiency)]
And another.
\end{proof}">|;