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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">L1</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>let </b><font color="Maroon">L2</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="E1:84"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">L1</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1_1"/>
( <font color="Maroon">L1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">L2</font> or <font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1"><b>case </b></a><a NAME="E1:84_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">L1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">L2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</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="E3:84_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T5" target="_self">Th5</a></i>;<br/><b>thus </b><a NAME="E4:84_1_1_1"/>
<font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a>, <a class="ref" href="euclidlp.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2"><b>case </b></a><a NAME="E1:84_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x0</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="E3:84_1_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T21" target="_self">Th21</a>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:84_1_1_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">L1</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  &amp; <font color="Maroon">L2</font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 <b>by </b><i><a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, <a class="txt" href="euclidlp.html#E15"><i><font color="Green">Lemma69</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">x1</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="E6:84_1_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/><b>consider </b><font color="Maroon">x2</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="E8:84_1_1_2"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T23" target="_self">Th23</a>, <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E9:84_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, <a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T25" target="_self">Th25</a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, <a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E10:84_1_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T25" target="_self">Th25</a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, </i>;<br/><a NAME="E11:84_1_1_2"/>
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1_1_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:84_1_1_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, <a class="ref" href="euclidlp.html#T29" target="_self">Th29</a>, <a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E4:84_1_1_2_1"/>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T30" target="_self">Th30</a>, <a class="ref" href="euclidlp.html#T28" target="_self">Th28</a>, </i>;<br/>
<a NAME="E5:84_1_1_2_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font>,<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="euclidlp.html#NR4" target="_self">are_ldependent2</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:84_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="euclidlp.html#T27" target="_self">Th27</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:84_1_1_2"/>
<font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:84"/>
<font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font>
 ;<br/>


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