<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>;<br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E43">A1</font></i>: 
( <font color="Maroon" title="c2">x1</font>,<font color="Maroon" title="c3">x2</font> <a href="euclidlp.html#NR4" title="EUCLIDLP:NR.4">are_ldependent2</a>  &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">a2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:47"/><i><font color="Green" title="E44">A2</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; ( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <b>by </b><i><a class="ref" href="euclidlp.html#D2" target="_self" title="EUCLIDLP:def.2">Def2</a></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" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="euclidlp.html#E3:47"><i><font color="Green" title="E44">A2</font></i></a></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" title="E45">A3</font></i>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><i><font color="Green" title="E46">A4</font></i>: <font color="Maroon" title="c4">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E3:47"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="euclidlp.html#T11" target="_self" title="EUCLIDLP:th.11">Th11</a></i>
<br/>.= 
 <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T54" title="RVSUM_1:th.54">RVSUM_1:54</a></i>
;<br/><i><font color="Green" title="E47">A5</font></i>: <font color="Maroon" title="c2">x1</font> = 
1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font>
<b>by </b><i><a class="txt" href="euclidlp.html#E1:47_1_1_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self" title="EUCLIDLP:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E2:47_1_1_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="euclidlp.html#T8" target="_self" title="EUCLIDLP:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font>
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self" title="EUCLIDLP:th.1">Th1</a></i>
;<br/><b>set </b><font color="Maroon" title="c6">t</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">a1</font>;<br/><b>take </b><font color="Maroon" title="c7">t</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">a1</font>;<br/><b>thus </b><a NAME="E4:47_1_1_1"/>
<font color="Maroon" title="c2">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:47"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="euclidlp.html#E3:47_1_1_1"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</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" title="E45">A6</font></i>: 
<font color="Maroon" title="c5">a2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><i><font color="Green" title="E46">A7</font></i>: <font color="Maroon" title="c5">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E3:47"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="euclidlp.html#T11" target="_self" title="EUCLIDLP:th.11">Th11</a></i>
<br/>.= 
 <a href="euclid.html#K6" title="EUCLID:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T54" title="RVSUM_1:th.54">RVSUM_1:54</a></i>
;<br/><i><font color="Green" title="E47">A8</font></i>: <font color="Maroon" title="c3">x2</font> = 
1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font>
<b>by </b><i><a class="txt" href="euclidlp.html#E1:47_1_1_2"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self" title="EUCLIDLP:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E2:47_1_1_2"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="euclidlp.html#T8" target="_self" title="EUCLIDLP:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">a2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x1</font>
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self" title="EUCLIDLP:th.1">Th1</a></i>
;<br/><b>set </b><font color="Maroon" title="c6">s</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">a2</font>;<br/><b>take </b><font color="Maroon" title="c7">s</font> = <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c4">a1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c5">a2</font>;<br/><b>thus </b><a NAME="E4:47_1_1_2"/>
<font color="Maroon" title="c2">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:47"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="euclidlp.html#E3:47_1_1_2"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</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" title="c2">x1</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c3">x2</font>
 ;<br/>


</div>
