<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font>;<br/>







<b>assume </b><a NAME="E1:73"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:73"/>
<font color="Maroon">L</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">n</font>
 
;<br/>
<b>then consider </b><font color="Maroon">L0</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E4:73"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L0</font> &amp;  ex <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> st <font color="Maroon">L0</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Olive">y1</font>,<font color="Olive">y2</font> )
 ;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">y2</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:73"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E7:73"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T12" target="_self">Th12</a>, <a class="txt" href="euclidlp.html#E15"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="euclid_4.html#T12">EUCLID_4:12</a></i>;<br/>
<a NAME="E8:73"/>
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T12" target="_self">Th12</a>, <a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:73"/><b>then </b>
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E10:73"/>
(  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 <b>by </b><i><a class="ref" href="euclidlp.html#T12" target="_self">Th12</a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>


</div>
