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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">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>assume </b><a NAME="E1:59"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">x</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>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:59"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="jordan2c.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:59"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E6:59"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1"/>
( <font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>suppose </b></a><a NAME="E1:59_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">g</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E3:59_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E4:59_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><i><font color="Green">E63</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="jordan2c.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><b>then </b><i><font color="Green">E64</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> = 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E4:59"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>

;<br/><b>then reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">g</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i>, <a class="txt" href="jordan2c.html#E1:59"><i><font color="Green">E47</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:59_1_1_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_3_1"/>
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y2</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E3:59_1_1"><i><font color="Green">E60</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_3_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_3_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , <a class="txt" href="jordan2c.html#E1:59_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:59_1_1_3_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:59_1_1_3_1_1"/><i><font color="Green">E131</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:59_1_1_3_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:59_1_1_3_1_1"/><i><font color="Green">E154</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E7:59_1_1_3_1_1"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E8:59_1_1_3_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><i><font color="Green">E158</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i2</font>

;<br/><a NAME="E10:59_1_1_3_1_1"/>
( 1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E6:59_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E1:59_1_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan2c.html#E1:59_1_1_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan2c.html#E2:59_1_1_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E12:59_1_1_3_1_1"/><b>then </b>
1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan2c.html#E4:59_1_1"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan2c.html#E5:59_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E13:59_1_1_3_1_1"/><b>then </b><i><font color="Green">E159</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E5:59_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E14:59_1_1_3_1_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:59_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E6:59_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E15:59_1_1_3_1_1"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:59_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E6:59_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E16:59_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><b>hence </b><a NAME="E17:59_1_1_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_3_1_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_3_1_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , <a class="txt" href="jordan2c.html#E1:59_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:59_1_1_3_1_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:59_1_1_3_1_2"/><i><font color="Green">E164</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:59_1_1_3_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:59_1_1_3_1_2"/><i><font color="Green">E166</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E7:59_1_1_3_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E8:59_1_1_3_1_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><i><font color="Green">E168</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">i2</font>

;<br/><a NAME="E10:59_1_1_3_1_2"/>
( 1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E6:59_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E1:59_1_1_3"><i><font color="Green">E65</font></i></a>, , , <a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E12:59_1_1_3_1_2"/><b>then </b>
1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan2c.html#E1:59_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E4:59_1_1"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan2c.html#E5:59_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E13:59_1_1_3_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E5:59_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E14:59_1_1_3_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:59_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E4:59_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E6:59_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E15:59_1_1_3_1_2"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:59_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E6:59_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E16:59_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><b>hence </b><a NAME="E17:59_1_1_3_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:59_1_1"/>
 ex <font color="Olive">y</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> st <br/>for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( not <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Olive">y</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2"><b>suppose </b></a><a NAME="E1:59_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:59_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E4:59_1_2"/><i><font color="Green">E172</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E5:59_1_2"/><i><font color="Green">E205</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="jordan2c.html#T7" target="_self">Th7</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:59_1_2"/><i><font color="Green">E206</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E7:59_1_2"/><i><font color="Green">E207</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then </b><i><font color="Green">E208</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1

;<br/><a NAME="E9:59_1_2"/><b>then </b><i><font color="Green">E209</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="jordan2c.html#T7" target="_self">Th7</a>, <a class="txt" href="jordan2c.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E10:59_1_2"/><i><font color="Green">E210</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="jordan2c.html#T7" target="_self">Th7</a>, <a class="txt" href="jordan2c.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><i><font color="Green">E211</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E4:59"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>

;<br/><b>then reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">g</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i>, <a class="txt" href="jordan2c.html#E1:59"><i><font color="Green">E47</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:59_1_2_3"/><i><font color="Green">E212</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_2_3_1"/>
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y2</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:59_1_2_3_1_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:59_1_2_3_1_1"/><i><font color="Green">E216</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i>, , , <a class="ref" href="jordan3.html#T17">JORDAN3:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i>, <a class="txt" href="jordan2c.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="jordan3.html#T32">JORDAN3:32</a></i>
;<br/><a NAME="E6:59_1_2_3_1_1"/><b>then </b>
1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:59_1_2_3_1_1"/><b>then </b><i><font color="Green">E217</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E8:59_1_2_3_1_1"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:59_1_2_3_1_1"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><b>hence </b><a NAME="E11:59_1_2_3_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:59_1_2_3_1_2"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_2_3_1_2"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:59_1_2_3_1_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:59_1_2_3_1_2"/><i><font color="Green">E223</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i>, , , <a class="ref" href="jordan3.html#T17">JORDAN3:17</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i2</font>
<b>by </b><i>, <a class="txt" href="jordan2c.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan2c.html#E6:59"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="jordan3.html#T32">JORDAN3:32</a></i>
;<br/><a NAME="E6:59_1_2_3_1_2"/><b>then </b>
1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:59_1_2_3_1_2"/><b>then </b><i><font color="Green">E224</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E8:59_1_2_3_1_2"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan2c.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:59_1_2_3_1_2"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><b>hence </b><a NAME="E11:59_1_2_3_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:59_1_2"/>
 ex <font color="Olive">y</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> st <br/>for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( not <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Olive">y</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
