<?xml version="1.0"?>
<div class="add">

<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">y1</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">x1</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">x2</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">x3</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">y2</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>assume </b><a NAME="E1:97"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</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">y2</font> <a href="hidden.html#R2">in</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> )
 ;<br/>

<b>then consider </b><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="E3:97"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1'</font> &amp;  ex <font color="Olive">a11</font>, <font color="Olive">a12</font>, <font color="Olive">a13</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">a11</font> <a href="real_1.html#K3">+</a> <font color="Olive">a12</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a13</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a11</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"><font color="Olive">a12</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Olive">a13</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span> ) )
 ;<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><b> such that </b><br/><a NAME="E5:97"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2'</font> &amp;  ex <font color="Olive">a21</font>, <font color="Olive">a22</font>, <font color="Olive">a23</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">a21</font> <a href="real_1.html#K3">+</a> <font color="Olive">a22</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">a23</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y2'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a21</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"><font color="Olive">a22</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Olive">a23</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="euclidlp.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E6:97"/>
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:97_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:97_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y2</font></span>)</span>
 ;<br/>
<b>consider </b><font color="Maroon">a11</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a12</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a13</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:97_1"/><i><font color="Green">E90</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a11</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a12</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a13</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a11</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"><font color="Maroon">a12</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a13</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T31" target="_self">Th31</a></i>;<br/>
<b>consider </b><font color="Maroon">a21</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a22</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a23</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:97_1"/><i><font color="Green">E105</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a21</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a22</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a23</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y2'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a21</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"><font color="Maroon">a22</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a23</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T32" target="_self">Th32</a></i>;<br/>
<i><font color="Green">E106</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a11</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a12</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a13</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a21</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><font color="Maroon">a22</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">a23</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euclidlp.html#E5:97"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E3:97_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="euclidlp.html#T31" target="_self">Th31</a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a>, <a class="ref" href="euclidlp.html#T33" target="_self">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a11</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a12</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a13</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a21</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a22</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a23</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a11</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a21</font></span>)</span></span>)</span> <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a12</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a22</font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a13</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a23</font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span>
<b>by </b><i/>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a11</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a21</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a12</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a22</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">a13</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a23</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a11</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a12</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a13</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a21</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a22</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a23</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font>
<b>by </b><i><a class="txt" href="euclidlp.html#E5:97"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E3:97_1"><i><font color="Green">E89</font></i></a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E10:97_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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#E5:97_1"><i><font color="Green">E90</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:97"/>
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</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>
 ;<br/>


</div>
