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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="E1:121"/><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> &amp; <font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L2</font> )
 ;<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="E3:121"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:121"/><i><font color="Green">E88</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#T40" target="_self">Th40</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">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="E6:121"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon">x3</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="E8:121"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E9:121"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">L1</font> <a href="hidden.html#R1">=</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#T41" target="_self">Th41</a>, , <a class="txt" href="euclidlp.html#E3:121"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E10:121"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="txt" href="euclidlp.html#E3:121"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E11:121"/>
<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T40" target="_self">Th40</a>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , , <a class="ref" href="euclidlp.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E12:121"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">P</font> =  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> as   <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> ;<br/>
<a NAME="E14:121"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T27" target="_self">Th27</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E15:121"/>
( <font color="Maroon">P</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">n</font> &amp; <font color="Maroon">P</font> is <a href="euclidlp.html#V1" target="_self">being_plane</a> &amp; <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T31" target="_self">Th31</a>, <a class="txt" href="euclidlp.html#E10:121"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, <a class="txt" href="euclidlp.html#E12:121"><i><font color="Green">E107</font></i></a>, , <a class="txt" href="euclidlp.html#E9:121"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T29" target="_self">Th29</a></i>;<br/>


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