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    <h2>Tarski Formalization Project Archives</h2>
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    <h3> Extensionality and Perpendicularity
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Szmielew treats lines as sets, so she omits some simple lemmas that are required in a formal treatment.  These can be proved
quite easily as soon as perpendicularity is defined (which is in Chapter 8).

<?php
$descriptions["ExtCol2"] = "If \$c\$ and \$d\$ are distinct points on \$Line(a,b)\$, then every point on \$Line(c,d)\$ is also on \$Line(a,b)\$.";
$descriptions["ext1"] = "If \$Col(a,b,c)\$ and \$Col(a,b,d)\$ and \$ a \\neq b\$, then \$Col(a,c,d)\$.";
$descriptions["ExtPerp"] = "If \$ab\$ and \$cd\$ determine the same line and \$perpAt(a,b,x,p,q)\$ then \$perpAt(c,d,x,p,q)\$. <br>" .
            "Used in   8.24 as well as subsequent chapters.";
 $descriptions["ExtPerp2"] = "If \$p$ and \$q$ lie on \$Line(a,b)\$ and \$p \\neq q\$, then \$pq \\perp cd\$ implies \$ab \\perp cd\$.";
$descriptions["ExtPerp3"] = "If \$a,b,c,d\$ are distinct and \$ba \perp ac\$ and \$Col(a,c,d)\$ then \$ba \\perp ad\$.";
$descriptions["ExtPerp4"] = "\$ab \\perp pq\$ implies \$ab \\perp qp\$.";
$descriptions["ExtPerp5"] = "If \$p$ and \$q$ lie on \$Line(a,b)\$ and \$p \\neq q\$, then \$ab \\perp cd\$ implies \$pq \\perp cd\$.";
$descriptions["ExtPerp6"] = "If \$p$ and \$q$ lie on \$Line(a,b)\$ and \$p \\neq q\$, then \$cd \\perp pq \$ implies \$cd \\perp ab\$.";
$Chapter = "8A";
include("DisplayResults.php");
?>