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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">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>let </b><font color="Maroon">L0</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">P</font> be    <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>;<br/>











<b>assume </b><a NAME="E1:124"/><i><font color="Green">E59</font></i>: 
( <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> &amp; <font color="Maroon">L1</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<b>assume </b><a NAME="E2:124"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">L0</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L0</font> )
 ;<br/>

<a NAME="E3:124"/>
<font color="Maroon">L1</font>,<font color="Maroon">L2</font> <a href="euclidlp.html#R7" target="_self">are_coplane</a> 
 
<b>by </b><i>, <a class="txt" href="euclidlp.html#E12:124"><i><font color="Green">E90</font></i></a></i>;<br/>
<a NAME="E4:124"/><b>then </b>
<font color="Maroon">L1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">L2</font>
 
<b>by </b><i>, <a class="ref" href="euclidlp.html#T41" target="_self">Th41</a></i>;<br/>
<b>then 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="E6:124"/><i><font color="Green">E88</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#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:124"/>
( <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="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="E9:124"/><i><font color="Green">E89</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">L2</font> )
 <b>by </b><i>, <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:124"/>
<font color="Maroon">L0</font> <a href="euclid_4.html#NR1" target="_self">is_line</a> 
 
<b>by </b><i>, <a class="ref" href="euclidlp.html#T13" target="_self">Th13</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="E12:124"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L0</font> )
 <b>by </b><i>, <a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E13:124"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">L0</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x2</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="txt" href="euclidlp.html#E16:124"><i><font color="Green">E106</font></i></a>, <a class="txt" href="euclidlp.html#E2:124"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E14:124"/>
<font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, , <a class="txt" href="euclidlp.html#E13:124"><i><font color="Green">E105</font></i></a>, <a class="txt" href="euclidlp.html#E16:124"><i><font color="Green">E106</font></i></a>, <a class="ref" href="euclidlp.html#T21" target="_self">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E16:124"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span> )
 <b>by </b><i/>;<br/>
<i><font color="Green">E107</font></i>: <font color="Maroon">x2</font> = 
<font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E19:124"><i><font color="Green">E108</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E18:124"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E19:124"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x</font>,<font color="Maroon">x1</font>,<font color="Maroon">x0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E20:124"/>
 <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x</font>,<font color="Maroon">x1</font>,<font color="Maroon">x0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="txt" href="euclidlp.html#E13:124"><i><font color="Green">E105</font></i></a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E21:124"/>
( <font color="Maroon">L0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">L0</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L1</font> )
 <b>by </b><i>, <a class="txt" href="euclidlp.html#E17:124"><i><font color="Green">E107</font></i></a>, , <a class="txt" href="euclidlp.html#E6:124"><i><font color="Green">E88</font></i></a>, , <a class="ref" href="euclidlp.html#T9" target="_self">Th9</a></i>;<br/>


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