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<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">x0</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="E1:59"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 ;<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="E3:59"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self">Th7</a>, <a class="ref" href="euclid_4.html#T14">EUCLID_4:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> or <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font>;<br/><b>thus </b><a NAME="E2:59_1_1_1"/>
( <font color="Maroon">y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E1:59"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font>;<br/><b>thus </b><a NAME="E2:59_1_1_2"/>
( <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E1:59"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:59"/>
 ex <font color="Olive">x1</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> )
 ;<br/>


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