Sindbad~EG File Manager

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<?php 
$Chapter = "5";
  $descriptions["Satz5.1"] = "Connectivity of betweenness <br>(Gupta 1965, one of Quaife's challenge problems)<br>" .
                           "\$a \\neq b \\land T(a,b,c) \\land T(a,b,d) \\rightarrow T(a,c,d) \\lor T(a,d,c)\$ <br>" .
                           "Wos has subsequently found a 96-step proof.";
  $descriptions["Satz5.2"] = "\$a \\neq b \\land T(a,b,c) \\land T(a,b,d) \\rightarrow T(b,c,d) \\lor T(b,d,c)   \$";
  $descriptions["Satz5.3"] = "\$ T(a,b,d) \\land T(a,c,d) \\rightarrow T(a,b,c) \\lor T(a,c,b)   \$";
  $descriptions["Satz5.5a"] = "left to right of Satz 5.5";
  $descriptions["Satz5.5b"] = "right to left of Satz 5.5";
 $descriptions["Satz5.6"] = "\$\\le\$ respects congruence";
 $descriptions["Satz5.7"] = "reflexivity of \$ \\le \$";
 $descriptions["Satz5.8"] = "transitivity of \$ \\le \$";
$descriptions["Satz5.9"] = "\$ ab \\le cd \\land cd \\le ef \\rightarrow ab \\le cd   \$";
$descriptions["Satz5.10"] = "\$ ab \\le cd  \\lor cd \\le ab    \$";
$descriptions["Satz5.11"] = "\$ aa \\le cd   \$";
$descriptions["Satz5.12a1"] = "\$Col(a,b,c) \\land T(a,b,c) \\rightarrow ab \\le ac \\land bc \\le ac    \$";
$descriptions["Satz5.12a2"] = "\$  Col(a,b,c) \\land T(a,b,c)\\rightarrow  bc \\le ac    \$";
$descriptions["NarbouxLemma1"] = "\$  T(a,b,c) \\land E(a,c,a,b)\\rightarrow c=b    \$ <br>".
                            " We used this to prove 5.12b (and almost certainly nowhere else). Narboux needed this in his formalization too.";
$descriptions["Satz5.12b"] = "\$ Col(a,b,c) \\land  ab \\le ac \\land bc \\le ac \\rightarrow T(a,b,c)   \$";
 

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