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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">y1</font> be    <a href="subset_1.html#M1">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="subset_1.html#M1">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="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>







<b>assume </b><a NAME="E1:55"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</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="E3:55"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y1</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">x1</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">x2</font></span>)</span>
 ;<br/>
<b>assume </b><a NAME="E4:55"/>
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:55"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">y2</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">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>
 ;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</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"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="euclid_4.html#E3:55"><i><font color="Green">E41</font></i></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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</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"><font color="Maroon">t</font> <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"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <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"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</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"><a href="euclid.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">t</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"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></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"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</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="euclid.html#K9">*</a> <font color="Maroon">x2</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">s</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></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"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <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"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">s</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></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"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <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"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span></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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">s</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">1 <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s</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="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></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> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <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">x2</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>take </b><font color="Maroon">a</font> = <font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font>;<br/>

<b>thus </b><a NAME="E8:55"/>
<font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</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">x2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid_4.html#T2" target="_self">Th2</a></i>;<br/>


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