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    <h2>Tarski Formalization Project Archives</h2>
    <h3> Chapter 9:  Sides of a line, and outer Pasch from inner Pasch.
    </h3>
    <p>There are two major results in this chapter:  the opposite-sides theorem (related to the idea that a line
    divides a plane into two regions), and the fact that inner Pasch implies outer Pasch.  Both these results,
    like the main results of Chapters 7 and 8,  are due to Gupta (1965).
    </p>
    <p> Hilbert and Tarski agree that "p and q are on the opposite side of Line(a,b)"  means that there is a point on the line
        strictly between p and q.  
    </p>
    <p>  Tarski defined "p and q are on the
    same side of line L" to mean that there is some point c such that both segments pc and qc meet
    line L, and none of p,q,c lie on L.  Hilbert just said it meant that pq does not meet L.  Tarski's
    definition has the advantage of working in n-space.   It also allows "plane" to be a defined concept: Three
    non-collinear points a,b,c  determine the plane consisting of all points that are on the same side of Line(a,b)
    as c,  or on the opposite side.   But then, are the points on same side as c all on the opposite side from 
    any one point on the opposite side?   Yes, they are,  according to Satz 9.8, and moreover, being on the 
    same side of a given line is a transitive relation (Satz 9.13).  
    </p>
    
    <p> These results assert the existence of certain intersection points of lines, and hence they are 
    intimately bound up with Pasch's axiom, which is the fundamental principle that a line entering a triangle
    (in the same plane as the triangle) must also exit the triangle.
    There are two forms of Pasch's axiom that are considered in Tarski's geometry:  inner Pasch, and outer Pasch.
    (See my paper <a href="http://www.michaelbeeson.com/research/papers/ConstructiveTarski.pdf"> "A constructive version of Tarski's geometry" </a> for diagrams.)
    </p>
     <p>  Both are good choices for an axiom, since 
     their logical form is simple and they do not rely on an upper dimension axiom to be true, i.e. they are 
     true in n-space as well as in the plane.  Tarski originally took outer Pasch as an axiom, since it was known 
     that outer Pasch implies inner Pasch.  Gupta showed that inner Pasch implies outer Pasch, which is Satz 9.6.
     After that, Szmielew took inner Pasch as an axiom (as is done in SST), although Tarski argued in a famous letter
     that the development might be easier with outer Pasch as an axiom.
     </p>
     <p> The theorem that inner Pasch implies outer Pasch rests on the opposite-side theorem Satz 9.5; its fairly short proof uses nothing else
        from Chapters 7,8, or 9.  Once both forms of Pasch are available, the derivation of the properties of "sameside"
        is not difficult.
     </p>The opposite-side theorem (Satz 9.5)  rests on what SST calls Lemma 9.4.   That "Lemma" gave Otter a 
     lot of trouble, so much that we upgraded it from "Lemma" to "Satz".  We had to break it into three cases and 
       prove them separately, and then derive Satz 9.4 from the three separately-proved cases,  and even that was
       not trivial.  
       
<?php
$Chapter = "9";
$descriptions["Satz9.2"] = "If \$a\$ and \$b<\$ are on opposite sides of line \$L\$, then
        \$b\$ and \$a\$ are on opposite sides of \$L\$.";
$descriptions["Satz9.3a"] = "The case of Satz 9.3 in which \$T(r,b,a)\$.  We had to prove this separately, and then use it to prove Satz 9.3.";
$descriptions["SideReflect"] = "\$ sameside(a,b,c) \\rightarrow sameside(s(m,a),s(m,b),s(m,c))\$. <br>" .
 "This is used in the proof of Satz 9.4, but Szmielew fails to state or prove it.";
$descriptions["Satz9.3"] = "called Lemma 9.3 in the book, used for 9.4.
        <br> If \$b\$ and \$a\$ are on opposite sides of line \$L\$, and the midpoint of \$ac\$ lies on line \$L\$, and \$r\$ is on \$L\$, and \$b\$ is a point on the ray from \$r\$ through \$a\$, then \$b\$ and \$c\$ are on opposite sides of \$L\$.";
$descriptions["Satz9.4a"] = "Satz 9.4 (see below),  with assumptions \$r\\neq s\$ and \$ac \\le ra\$, 
            and conclusion \$sameside(u,r,a) -> sameside(s(m,u),s,c)\$";
$descriptions["Satz9.4a2"] = "Satz 9.4 (see below), with assumptions \$r\\neq s\$ and \$ac \\le ra \$,  and conclusion  \$sameside(s(m,u),s,c) -> sameside(u,r,a)\$";
$descriptions["Satz9.4b"] = "Satz 9.4 assuming \$r=s\$.";
$descriptions["Satz9.4"] = "First part of Lemma 9.4 in the book (line 1 implies left-to-right of line 2), here derived from its cases 9.4a, 9.4a2, and 9.4b.";
$descriptions["Satz9.4c"] = "Second half of Lemma 9.4 in the book (line 1 implies line 3). <br>" .
           "Assumes \$sameside(u,r,a)\$ and \$sameside(v,s,c)\$, and concludes \$opposite(u,p,q,v)\$, where \$p,q\$ give the line called \$A\$ in the book.";
$descriptions["Satz9.5"] = "Opposite sides theorem: if \$a\$a and \$ b\$ are on the same side of point \$r\$ (i.e., on the same ray from \$r\$), and \$a\$ and \$c\$ are on opposite sides of line \$L\$, then \$b\$ and \$c\$ are on opposite sides of \$L\$.";

$descriptions["Satz9.6"] = "Inner Pasch implies outer Pasch (Gupta 1965), <br>" .
                      "This was one of Quaife's challenge problems.";
$descriptions["Satz9.8"] = "If \$a\$ and \$c\$ are on opposite sides of line \$L\$, and \$a\$ and \$b\$ are on the same side of \$L\$,  then \$b\$ and \$c\$ are on opposite sides of \$L\$ .";
$descriptions["Satz9.9"] = "If \$a\$ and \$b\$ are on opposite sides of line \$L\$, then  \$a\$ and \$b\$ are not on the same side of \$L\$ .";
$descriptions["Satz9.13"] = "\$samesideline\$ is a transitive relation.";
$descriptions["Satz9.16"] = "If \$a\$ and \$b\$ are on the same side of  \$Line(p,q)\$, and \$a\$ and \$r\$ are on the same ray emanating from \$q\$,  then \$r\$ and \$b\$ are on the same side of \$Line(p,q)\$. <br>".
 "This theorem is not in the book, but we needed it for Satz 11.15b. (The number 9.16
             is used for a remark in the book.)";
$descriptions["Satz9.19b"] = "If \$p\$ is on line \$L\$, and \$a\$ and \$b\$ are on the same ray emanating from \$p\$ and \$a\$ is not on \$L\$, then \$a\$ and \$b\$ are on the same side of \$L.\$" .
"This is the right-to-left part of Satz 9.19.";
 


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