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    <h2>Tarski Formalization Project Archives</h2>
    <p></p>   
    <h3>Rays,  and  comparison of segments</h3> 
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<p> $sameside(a,b,c)$ means that $a$  and  $c$ are on the same side of point $b$, i.e. on the same ray emanating from $b$.
By definition that means $T(b,a,c)\lor T(b,c,a)$. 

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$Chapter = "6";
$descriptions["Satz6.2a"]="\$a\\neq b \\land  b\\neq p \\land  c\\neq p \\land  T(a,p,c) \\land  T(b,p,c) \\rightarrow  sameside(a,p,b)\$";
$descriptions["Satz6.2b"]="\$ a\\neq b \\land  b\\neq p \\land  c\\neq p \\land  T(a,p,c) \\land  sameside(a,p,b)\\rightarrow T(b,p,c) \$";
$descriptions["Satz6.3a"]="\$sameside(a,p,b)\\rightarrow  a\\neq p \\land  b \\neq p \\land \\exists c(c \\neq p \\land  T(a,p,c) \\land  T(b,p,c)) \$";
$descriptions["Satz6.3b"]="\$a\\neq p \\land  b \\neq p \\land \\exists c(c \\neq p \\land  T(a,p,c) \\land  T(b,p,c))   \\rightarrow  sameside(a,p,b)\$";
$descriptions["Satz6.4a"]="\$ sameside(a,p,b) \\rightarrow  Col(a,p,b) \\land\\neg T(a,p,b)  \$";
$descriptions["Satz6.4b"]="\$Col(a,p,b) \\land \\neg T(a,p,b)  \\rightarrow  sameside(a,p,b)\$";
$descriptions["Satz6.5"]="(reflexivity) \$a\\neq p \\rightarrow  sameside(a,p,a) \$";
$descriptions["Satz6.6"]="(symmetry) \$sameside(a,p,b) \\rightarrow  sameside(b,p,a)\$";
$descriptions["Satz6.7"]="(transitivity) \$sameside(a,p,b) \\land  sameside(b,p,c) \\rightarrow  sameside(a,p,c)\$";
$descriptions["Satz6.11a"]="\$r\\neq a \\land  b\\neq c \\rightarrow  \\exists x(sameside(x,a,r) \\land  E(a,x,b,c)) \$ ";
$descriptions["Satz6.11b"]="uniqueness of the \$x\$ in 6.11a";
$descriptions["Satz6.13a"]="\$sameside(a,p,b) \\land  pa \\le pb \\rightarrow  T(p,a,b)\$";
$descriptions["Satz6.13b"]="\$sameside(a,p,b) \\land  T(p,a,b) \\rightarrow  pa \\le pb \$";
$descriptions["Satz6.15a"]=" \$ p\\neq q \\land  p\\neq r \\land  T(q,p,r) \\land  Col(p,q,a) \\rightarrow sameside(a,p,q) or a=p or sameside(a,p,r)\$";
$descriptions["Satz6.15b"]="\$p\\neq q \\land  p\\neq r \\land  T(q,p,r) \\land  sameside(a,p,q) \\rightarrow  Col(p,q,a) \$";
$descriptions["Satz6.15c"]=" \$p\\neq q \\land  p\\neq r \\land  T(q,p,r) \\land  sameside(a,p,r) \\rightarrow  Col(p,q,a) \$";
$descriptions["Satz6.15d"]="\$p\\neq q \\land  p\\neq r \\land  T(q,p,r) \\land  a=p \\rightarrow  Col(p,q,a) \$ ";
$descriptions["Satz6.16a"]="\$p\\neq q \\land  p\\neq r \\land  T(q,p,r) \\land  a=p \\rightarrow  Col(p,q,a) \$ ";
$descriptions["Satz6.16b"]="\$p\\neq q \\land  s\\neq p \\land  Col(p,q,s) \\land  Col(p,q,x) \\rightarrow   Col(p,s,x)\$";
$descriptions["Satz6.17a"]=" \$p\\neq q \\rightarrow  Col(p,q,p)\$";
$descriptions["Satz6.17b"]="\$ p\\neq q \\land  Col(p,q,x) \\rightarrow  Col(q,p,x) \$";
$descriptions["Satz6.18"]="\$ a\\neq b \\land  p\\neq q \\land  Col(p,q,a) \\land  Col(p,q,b) \\land  Col(p,q,x) \\rightarrow  Col(a,b,x)\$. 
        <br> This is a special case of Satz 6.21, used in proving Satz 6.21.";
$descriptions["Satz6.21"]="If \$Line(p,q)\$ and  \$Line(a,b)\$ have two different points in common then they coincide. ";
$descriptions["Satz6.25"]="There exists a point \$x\$ not on \$Line(p,q)\$, that is, such that not \$Col(p,q,x)\$. ";
$descriptions["Satz6.28"]="If \$sameside(a,b,c)\$  and  \$sameside(a1,b1,c1)\$, and  \$ba\$ and  \$bc\$ are respectively
        congruent to \$b_1a_1\$ and  \$b_1c_1\$, then \$ac\$ is congruent to \$a_1c_1\$. <br>" .
        "This is used in Satz 11.4, but is never proved in the book, and  belongs in Chapter 6,
        so we give it the name Satz 6.28.";


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