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    <h2>Tarski Formalization Project Archives</h2>
    
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$Chapter = "2";

$descriptions["Satz2.1"]= "(reflexivity) \$E(a,b,a,b)\$";
$descriptions["Satz2.2"]= "(symmetry) \$E(a,b,c,d) \\rightarrow E(c,d,a,b)\$";
$descriptions["Satz2.3"]= "(transitivity) \$E(a,b,c,d) \\land E(c,d,e,f) \\rightarrow E(a,b,e,f)\$";
$descriptions["Satz2.4"]= "\$E(a,b,c,d) \\rightarrow E(b,a,c,d)\$";
$descriptions["Satz2.5"]= "\$E(a,b,c,d) \\rightarrow E(a,b,d,c)\$";
$descriptions["Satz2.8"]= "\$E(a,a,b,b)\$  (all null segments are congruent)";
$descriptions["Satz2.11"]= "\$T(a,b,c,) \\land T(a1,b1,c1) \\land  E(a,b,a1,b1) \\land E(b,c,b1,c1) \\rightarrow E(a,c,a1,c1)\$";
$descriptions["Satz2.12"]= "(uniqueness of segment extension) <br>" .
    " \$ q!=a \\land T(q,a,x) \\land E(a,x,b,c) \\land    T(q,a,y) \\land E(a,y,b,c) \\rightarrow x=y\$";
$descriptions["Satz2.13"]= "\$ E(x,y,u,u) \\rightarrow x=y.\$  Anything congruent to a null segment is a null segment." .
      "This theorem does not occur in the book, and we don't actually need it, but it seems pretty basic.";
$descriptions["Satz2.14"]= "\$E(x,y,u,v) \\rightarrow E(y,x,v,u)\$.
        This theorem does not occur in the book, but we need it in Satz 10.12a and Satz 11.15b.";
$descriptions["Satz2.15"]= "\$T(a,b,c) \\land T(A,B,C) \\land E(a,b,B,C) \\land E(b,c,A,B) \\rightarrow E(a,c,A,C)\$. <br>" .
"This theorem, very similar to 2.11, does not occur in the book, but we need it in Satz 11.15b.
            It can be proved very easily using Satz 3.1, but here we prove it without Satz 3.1, which is not quite trivial.";
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