# LaTeX2HTML 2008 (1.71) # Associate images original text with physical files. $key = q/{displaymath}A^{-1}ClongleftrightarrowC'A'^{-1}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="128" HEIGHT="24" BORDER="0" SRC="|."$dir".q|img36.png" ALT="\begin{displaymath} A^{ -1} C \longleftrightarrow C' A'^{ -1} \end{displaymath}">|; $key = q/P_1^{-1};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="34" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img35.png" ALT="$P_1^{ -1}$">|; $key = q/R_{c};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img4.png" ALT="$R_{c}$">|; $key = q/==;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="29" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" SRC="|."$dir".q|img28.png" ALT="$==$">|; $key = q/R_l;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img11.png" ALT="$R_l$">|; $key = q/{displaymath}P_2parallelP_1Longrightarrow{P_2}'P_1longleftrightarrow{P_1}'P_2.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="205" HEIGHT="28" BORDER="0" SRC="|."$dir".q|img31.png" ALT="\begin{displaymath}P_2 \parallel P_1 \Longrightarrow {P_2}' P_1 \longleftrightarrow {P_1}' P_2. \end{displaymath}">|; $key = q/R_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img7.png" ALT="$R_1$">|; $key = q/P^{-1};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" SRC="|."$dir".q|img20.png" ALT="$P^{ -1}$">|; $key = q/P_2'P_1longleftrightarrowP_1'P_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="112" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img32.png" ALT="$ P_2' P_1 \longleftrightarrow P_1' P_2 $">|; $key = q/R_1bigcapR_2bigcapR_3ldots;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="121" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img13.png" ALT="$ R_1 \bigcap R_2 \bigcap R_3 \ldots$">|; $key = q/P_2';MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img26.png" ALT="$P_2'$">|; $key = q/P;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" SRC="|."$dir".q|img21.png" ALT="$P$">|; $key = q/P_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img17.png" ALT="$P_2$">|; $key = q/+;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img3.png" ALT="$+$">|; $key = q/P_1'^{-1};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img34.png" ALT="$P_1'^{ -1}$">|; $key = q/f_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img5.png" ALT="$f_1$">|; $key = q/P_2P_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img18.png" ALT="$P_2P_1$">|; $key = q/ddotfrown;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0" SRC="|."$dir".q|img29.png" ALT="$\ddot\frown$">|; $key = q/P_1parallelP_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img19.png" ALT="$P_1\parallel P_2$">|; $key = q/<;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img1.png" ALT="$<$">|; $key = q/P_1';MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="22" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img25.png" ALT="$P_1'$">|; $key = q/R_1bigcupR_2bigcupR_3ldots;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="120" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img12.png" ALT="$ R_1 \bigcup R_2 \bigcup R_3 \ldots$">|; $key = q/R_1backslash(R_2bigcupR_3bigcupldots;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="132" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img15.png" ALT="$ R_1 \backslash (R_2 \bigcup R_3 \bigcup \ldots$">|; $key = q/>;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img2.png" ALT="$>$">|; $key = q/R_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img9.png" ALT="$R_2$">|; $key = q/R_p;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img10.png" ALT="$R_p$">|; $key = q/{displaymath}P_2P_1longleftrightarrow{P_1}'{P_2}'.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="119" HEIGHT="26" BORDER="0" SRC="|."$dir".q|img24.png" ALT="\begin{displaymath}P_2 P_1 \longleftrightarrow {P_1}' {P_2}'. \end{displaymath}">|; $key = q/(P^{-1}P);MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" SRC="|."$dir".q|img22.png" ALT="\( P^{ -1} P \)">|; $key = q/R_c;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img6.png" ALT="$R_c$">|; $key = q/f_2;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img8.png" ALT="$f_2$">|; $key = q/longleftrightarrow;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="34" HEIGHT="15" ALIGN="BOTTOM" BORDER="0" SRC="|."$dir".q|img27.png" ALT="$\longleftrightarrow $">|; $key = q/P_1'^{-1}P_2'longleftrightarrowP_2P_1^{-1};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="140" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img33.png" ALT="$ P_1'^{ -1} P_2' \longleftrightarrow P_2 P_1^{ -1} $">|; $key = q/(Longrightarrow);MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="30" HEIGHT="13" ALIGN="BOTTOM" BORDER="0" SRC="|."$dir".q|img30.png" ALT="\( \Longrightarrow \)">|; $key = q/{displaymath}(P_2P_1)^{-1}=P_1^{-1}P_2^{-1}.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="146" HEIGHT="28" BORDER="0" SRC="|."$dir".q|img23.png" ALT="\begin{displaymath}(P_2 P_1)^{ -1} = P_1^{ -1} P_2^{ -1}. \end{displaymath}">|; $key = q/SbackslashSrightarrowemptyset;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img14.png" ALT="$ S \backslash S \rightarrow \emptyset $">|; $key = q/P_1;MSF=1.6;AAT/; $cached_env_img{$key} = q|<IMG WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" SRC="|."$dir".q|img16.png" ALT="$P_1$">|; 1;