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    <h2>Tarski Formalization Project Archives</h2>
     
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$Chapter = "4";
$descriptions["Satz4.2"] = "\$ T(a,b,c) \\land T(a_1,b_1,c_1) \\land E(a,c,a_1,c_1) \\land E(b,c,b_1,c_1).
         \\land E(a,d,a_1,d1) \\land E(c,d,c_1,d1) \\rightarrow E(b,d,b_1,d1)\$<br>".
        "This is the ``inner five-segment theorem'', similar to the 5-segment axiom but 
        with the segments to be proved equal on the inside, rather than the outside.";
 
$descriptions["Satz4.3"] = "\$T(a,b,c) \\land T(a_1,b_1,c_1) \\land E(a,c,a_1,c_1)   \\land E(b,c,b_1,c_1) \\rightarrow E(a,b,a_1,b_1) \$";
$descriptions["Satz4.5"] = "\$T(a,b,c) \\land ac = a_1c_1 \\rightarrow \\exists z(T(a_1,z,c_1) \\land (a,b,c) \\cong (a_1,z,c_1) \$ <br>" .
"We give an explicit term \$insert(a,b,a_1,c_1)\$ that produces \$z\$, so a new Skolem function is not needed.";
$descriptions["Satz4.6"] = "\$ T(a,b,c) \\land E(a,c,a_1,c_1) \\rightarrow   \\exists b_1(T(a_1,b_1,c_1) \\land E3(a,b,c,a_1,b_1,c_1)) \$. <br>" .
"We also show that \$b_1\$ depends only on \$a,b,a_1,c_1\$ (not on \$c\$) by an explicit term. ";
$descriptions["Satz4.11a"] = "\$Col(a,b,c) \\rightarrow Col(b,c,a)  \$";
$descriptions["Satz4.11b"] = "similarly for all permutations of \$(a,b,c)\$";
$descriptions["Satz4.11c"] = "\$  \$";
$descriptions["Satz4.11d"] = "\$  \$";
$descriptions["Satz4.11e"] = "\$  \$";
$descriptions["Satz4.12"] = "\$Col(a,a,b)  \$";
$descriptions["Satz4.12b"] = "\$ Col(a,b,a) \$";
$descriptions["Satz4.13"] = "\$ Col(a,b,c) \\land E3(a,b,c,a_1,b_1,c_1) \\rightarrow Col(a_1,b_1,c_1) \$";
$descriptions["Satz4.14"] = "\$ Col(a,b,c) \\land E(a,b,a_1,b_1) \\rightarrow \\exists c_1(E3(a,b,c,a_1,b_1,c_1))  \$";
$descriptions["Satz4.16"] = "\$Col(a,b,c) \\land E3(a,b,c,a_1,b_1,c_1) \\land E(a,d,a_1,d1)
         \\land E(b,d,b_1,d1) \\land a\\neq b \\rightarrow E(c,d,c_1,d1)  \$";
$descriptions["Satz4.17"] = "\$ a\\neq b \\land Col(a,b,c) \\land E(a,p,a,q) \\land E(b,p,b,q) \\rightarrow E(c,p,c,q)  \$";
$descriptions["Satz4.18"] = "\$ a\\neq b \\land Col(a,b,c) \\land E(a,c,a,c_1) \\land E(b,c,b,c_1) \\rightarrow c = c_1   \$";
$descriptions["Satz4.19"] = "\$  T(a,c,b) \\land E(a,c,a,c_1) \\land E(b,c,b,c_1) \\rightarrow c = c_1  \$";

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?>