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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="E1:81"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">L</font> <a href="euclid_4.html#NR1" target="_self">is_line</a> 
 ;<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="E3:81"/><i><font color="Green">E66</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">L</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#T17" target="_self">Th17</a>, <a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span>;<br/>
<a NAME="E4:81"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:81"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, <a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>;<br/>
<a NAME="E6:81"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</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="E7:81"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <font color="Maroon">x</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T21" target="_self">Th21</a>, <a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, <a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">L0</font> =  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span></span>)</span> as    <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> <b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self">Th1</a></i>;<br/>
<b>take </b>
<font color="Maroon">L0</font>
;<br/>

<b>thus </b><a NAME="E9:81"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L0</font> &amp; <font color="Maroon">L0</font> <a href="euclidlp.html#R5" target="_self">//</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="euclid_4.html#T10">EUCLID_4:10</a>, <a class="ref" href="euclidlp.html#T18" target="_self">Th18</a>, <a class="ref" href="euclidlp.html#T23" target="_self">Th23</a>, <a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a></i>;<br/>


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