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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">x</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="E1:114"/><i><font color="Green">E59</font></i>: 
( not <font color="Maroon">x</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">P</font> being    <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><b> such that </b><br/><a NAME="E3:114"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<b>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>, <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="E5:114"/><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">x1</font>,<font color="Maroon">x2</font> &amp; <font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</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><a class="ref" href="euclidlp.html#T37" target="_self">Th37</a>, </i>;<br/>
<a NAME="E6:114"/>
( <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> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#D2" target="_self">Def2</a>, <a class="ref" href="euclid_4.html#T10">EUCLID_4:10</a></i>;<br/>
<a NAME="E7:114"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x</font>,<font color="Maroon">x2</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T38" target="_self">Th38</a>, <a class="ref" href="euclidlp.html#D2" target="_self">Def2</a>, <a class="ref" href="euclidlp.html#T33" target="_self">Th33</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E8:114"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> is <a href="euclidlp.html#V1" target="_self">being_plane</a> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T38" target="_self">Th38</a>, <a class="txt" href="euclidlp.html#E43"><i><font color="Green">Lemma98</font></i></a>, <a class="ref" href="euclidlp.html#D2" target="_self">Def2</a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a></i>;<br/>


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