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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="E1:112"/><i><font color="Green">E59</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>  &amp; <font color="Maroon">L1</font>,<font color="Maroon">L2</font> <a href="euclidlp.html#R7" target="_self">are_coplane</a>  )
 ;<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>, <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>, <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="E3:112"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E5:112"><i><font color="Green">E89</font></i></a></i>;<br/>
<b>set </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>;<br/>
<b>assume </b><a NAME="E4:112"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">L1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">L2</font>
 ;<br/>

<a NAME="E5:112"/><i><font color="Green">E89</font></i>: 
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K5" target="_self">plane_of_REAL</a> <font color="Maroon">n</font>
 
;<br/>
<b>consider </b><font color="Maroon">y0</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">y1</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="E7:112"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y0</font>,<font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T36" target="_self">Th36</a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<b>consider </b><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>, <font color="Maroon">y3</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="E9:112"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y3</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y2</font>,<font color="Maroon">y3</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T36" target="_self">Th36</a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<a NAME="E10:112"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#D2" target="_self">Def2</a>, <a class="txt" href="euclidlp.html#E43"><i><font color="Green">Lemma98</font></i></a>, <a class="ref" href="euclid_4.html#T10">EUCLID_4:10</a></i>;<br/>
<b>consider </b><font color="Maroon">y</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="E12:112"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y0</font>,<font color="Maroon">y1</font> &amp; <font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y</font> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E13:112"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">y</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#D2" target="_self">Def2</a>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, <a class="ref" href="euclidlp.html#T40" target="_self">Th40</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="euclidlp.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E14:112"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</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">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y</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#D2" target="_self">Def2</a>, , </i>;<br/>
<a NAME="E15:112"/>
<font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, </i>;<br/>
<a NAME="E16:112"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y</font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y'</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="E18:112"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T36" target="_self">Th36</a>, <a class="ref" href="euclid_4.html#T15">EUCLID_4:15</a></i>;<br/>
<a NAME="E19:112"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y</font>,<font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="euclid_4.html#T10">EUCLID_4:10</a></i>;<br/>
<a NAME="E20:112"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">y'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y</font>,<font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y</font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 
<b>by </b><i><a class="ref" href="euclidlp.html#D2" target="_self">Def2</a>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , , , </i>;<br/>
<a NAME="E21:112"/><i><font color="Green">E113</font></i>: 
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">y</font>,<font color="Maroon">y'</font>,<font color="Maroon">y2</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#D2" target="_self">Def2</a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a>, , <a class="ref" href="euclidlp.html#T38" target="_self">Th38</a>, <a class="ref" href="euclidlp.html#T40" target="_self">Th40</a>, <a class="txt" href="euclidlp.html#E9:112"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T33" target="_self">Th33</a></i>;<br/>
<b>take </b>
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
;<br/>

<b>thus </b><a NAME="E22:112"/>
(  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</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">L1</font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">L2</font> <a href="tarski.html#R1">c=</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp;  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</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="ref" href="euclidlp.html#D1" target="_self">Def1</a>, <a class="txt" href="euclidlp.html#E10:112"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E9:112"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a></i>;<br/>


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