<?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">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="E1:80"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">L1</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">L2</font> )
 ;<br/>

<b>assume </b><a NAME="E2:80"/><i><font color="Green">E66</font></i>: 
not <font color="Maroon">L1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">L2</font>
 ;<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>, <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>, <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="E4:80"/><i><font color="Green">E88</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> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font> &amp; <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T16" target="_self">Th16</a>, <a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:80"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="euclidlp.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E8:80"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</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">y1</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#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E9:80"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E10:80"/><b>then </b><i><font color="Green">E105</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#T18" target="_self">Th18</a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</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="E12:80"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> )
 <b>by </b><i><a class="ref" href="euclid_4.html#T15">EUCLID_4:15</a></i>;<br/>
<a NAME="E13:80"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, <a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T23" target="_self">Th23</a>, </i>;<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><b> such that </b><br/><a NAME="E15:80"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y0</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclid_4.html#T15">EUCLID_4:15</a></i>;<br/>
<a NAME="E16:80"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y0</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, <a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, </i>;<br/>
<a NAME="E17:80"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</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">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</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#T23" target="_self">Th23</a>, <a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, </i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E19:80"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">x0</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">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T23" target="_self">Th23</a>, <a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, <a class="ref" href="euclidlp.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E20:80"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T21" target="_self">Th21</a>, <a class="ref" href="euclidlp.html#T28" target="_self">Th28</a>, <a class="ref" href="euclidlp.html#T29" target="_self">Th29</a>, </i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E22:80"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, <a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, <a class="ref" href="euclidlp.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E23:80"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T21" target="_self">Th21</a>, <a class="ref" href="euclidlp.html#T28" target="_self">Th28</a>, <a class="ref" href="euclidlp.html#T31" target="_self">Th31</a>, </i>;<br/>
<a NAME="E24:80"/>
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y2</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T18" target="_self">Th18</a>, <a class="ref" href="euclidlp.html#T30" target="_self">Th30</a>, </i>;<br/>
<a NAME="E25:80"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T32" target="_self">Th32</a>, </i>;<br/>
<a NAME="E26:80"/><i><font color="Green">E152</font></i>: 
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">x0</font> <a href="tarski.html#R1">c=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:80_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">x0</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="E3:80_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">y</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">x</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">x0</font></span>)</span>
 ;<br/>
<i><font color="Green">E154</font></i>: <font color="Maroon">y</font> = 
<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"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></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> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E6:80"><i><font color="Green">E89</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></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> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x0</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <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"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:80_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">x0</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">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T33" target="_self">Th33</a>, </i>;<br/>
<font color="Maroon">y</font> = 
<font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a>, <a class="txt" href="euclidlp.html#E8:80"><i><font color="Green">E90</font></i></a>, <a class="ref" href="euclidlp.html#T35" target="_self">Th35</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y0</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y0</font></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"><a href="euclid.html#K6">-</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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y0</font></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"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <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">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y0</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E8:80_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y0</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E27:80"/><i><font color="Green">E156</font></i>: 
 <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y0</font> <a href="tarski.html#R1">c=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:80_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">y0</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="E3:80_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">y</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">x</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">y0</font></span>)</span>
 ;<br/>
<i><font color="Green">E158</font></i>: <font color="Maroon">y</font> = 
<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"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></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> <font color="Maroon">y0</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E8:80"><i><font color="Green">E90</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></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> <font color="Maroon">y0</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y0</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <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"><font color="Maroon">y0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:80_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">y0</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">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T33" target="_self">Th33</a>, </i>;<br/>
<font color="Maroon">y</font> = 
<font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a>, <a class="ref" href="euclidlp.html#T35" target="_self">Th35</a>, </i>
<br/>.= 
<font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x0</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</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/>.= 
<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"><a href="euclid.html#K6">-</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">a</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</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><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <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">t</font> <a href="real_1.html#K4">*</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/>
<b>hence </b><a NAME="E8:80_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<font color="Maroon">x0</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E28:80"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:80"><i><font color="Green">E88</font></i></a>, <a class="txt" href="euclidlp.html#E6:80"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, <a class="ref" href="euclidlp.html#T25" target="_self">Th25</a>, <a class="ref" href="euclidlp.html#T27" target="_self">Th27</a>, <a class="ref" href="euclidlp.html#T16" target="_self">Th16</a></i>;<br/>


</div>
