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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">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>assume </b><a NAME="E1:46"/>
<font color="Maroon">x1</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:46"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:46"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 ;<br/>

 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> = 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x1</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</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">1 <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"><font color="Maroon">r</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">1 <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> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:46"/>
contradiction
 <b>by </b><i>, </i>;<br/>


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