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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="E1:47"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="euclidlp.html#NR4" target="_self">are_ldependent2</a>  &amp; <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">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:47"/><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a1</font> <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">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; ( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>case </b></a><a NAME="E1:47_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><i><font color="Green">E89</font></i>: <font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="euclid.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E90</font></i>: <font color="Maroon">x1</font> = 
1 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K6">/</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font>
<b>by </b><i/>
;<br/><b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">a1</font>;<br/><b>take </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">a1</font>;<br/><b>thus </b><a NAME="E4:47_1_1_1"/>
<font color="Maroon">x1</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="euclidlp.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><b>case </b></a><a NAME="E1:47_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><i><font color="Green">E106</font></i>: <font color="Maroon">a2</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="euclid.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i/>
;<br/><i><font color="Green">E107</font></i>: <font color="Maroon">x2</font> = 
1 <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="real_1.html#K6">/</a> <font color="Maroon">a2</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">a2</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">a2</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">1 <a href="real_1.html#K6">/</a> <font color="Maroon">a2</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a1</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">a2</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font>
<b>by </b><i/>
;<br/><b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">a2</font>;<br/><b>take </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a1</font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">a2</font>;<br/><b>thus </b><a NAME="E4:47_1_1_2"/>
<font color="Maroon">x1</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="euclidlp.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<font color="Maroon">x1</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">x2</font>
 ;<br/>


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