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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x0</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c1">n</font>;<br/>





<b>assume </b><a NAME="E1:58"/>
( <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c3">L</font> <a href="euclid_4.html#NR1" title="EUCLID_4:NR.1">is_line</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">y1</font>, <font color="Maroon" title="c5">y2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:58"/><i><font color="Green" title="E51">A1</font></i>: 
( <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c5">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y2</font> )
 <b>by </b><i><a class="ref" href="euclid_4.html#T14" title="EUCLID_4:th.14">EUCLID_4:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1"/>
( <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> or <font color="Maroon" title="c4">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1"><b>case </b></a><a NAME="E1:58_1_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c5">y2</font>;<br/><b>thus </b><a NAME="E2:58_1_1_1"/>
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:58"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="euclidlp.html#E1:58_1_1_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2"><b>case </b></a><a NAME="E1:58_1_1_2"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x0</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y1</font>;<br/><b>thus </b><a NAME="E2:58_1_1_2"/>
( <font color="Maroon" title="c6">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:58"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="euclidlp.html#E1:58_1_1_2"><i><font color="Green" title="E52">A3</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="E5:58"/>
 ex <font color="Olive" title="b1">x1</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x0</font> &amp; <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">L</font> )
 ;<br/>


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