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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:60"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:60"/><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jordan2c.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E5:60"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E6:60"/><b>then </b><i><font color="Green">E57</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</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="60_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">c<sub>4</sub></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="60_1_1"><b>suppose </b></a><a NAME="E1:60_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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#K13">*&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">c<sub>3</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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="E3:60_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></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#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></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#K13">*&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">c<sub>3</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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="E4:60_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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#K13">*&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">E61</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">c<sub>3</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E6:60"><i><font color="Green">E57</font></i></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/><b>then </b><i><font color="Green">E62</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> = 
1 <a href="binop_2.html#K9">+</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E3:60_1_1"><i><font color="Green">E59</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">c<sub>3</sub></font>

;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan2c.html#T52" target="_self">Th52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:60_1_1_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_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="E1:60_1_1_3_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E1:60_1_1_3"><i><font color="Green">E63</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:60_1_1_3_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_1_1_3_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:60_1_1_3_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:60_1_1_3_1_1"/><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:60_1_1_3_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:60_1_1_3_1_1"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E7:60_1_1_3_1_1"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E8:60_1_1_3_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binarith.html#K5">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</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#K5">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</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">c<sub>4</sub></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="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>

;<br/><a NAME="E10:60_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">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_1"><i><font color="Green">E61</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">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>3</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E4:60_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_1_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_1_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_1_3_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan2c.html#E8:60_1_1_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E9:60_1_1_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E12:60_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">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_1_3_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E2:60_1_1_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_1_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E13:60_1_1_3_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_1_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E14:60_1_1_3_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60_1_1_3_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_1_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E15:60_1_1_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:60_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_1_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E16:60_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E13:60_1_1_3_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan2c.html#E14:60_1_1_3_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E17:60_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E18:60_1_1_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:60_1_1_3_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_1_1_3_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:60_1_1_3_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:60_1_1_3_1_2"/><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:60_1_1_3_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:60_1_1_3_1_2"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E7:60_1_1_3_1_2"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E8:60_1_1_3_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="binarith.html#K5">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</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#K5">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</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">c<sub>4</sub></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="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>

;<br/><a NAME="E10:60_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">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_1"><i><font color="Green">E61</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">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>3</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E4:60_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_1_3_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_1_3_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_1_3_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan2c.html#E8:60_1_1_3_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan2c.html#E9:60_1_1_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E12:60_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">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_1_3_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E2:60_1_1_3_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_1_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E13:60_1_1_3_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_1_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E14:60_1_1_3_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:60_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_1_3_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_1_3_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E15:60_1_1_3_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:60_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_1_3_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E16:60_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E13:60_1_1_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan2c.html#E14:60_1_1_3_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E17:60_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E18:60_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="E9:60_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2"><b>suppose </b></a><a NAME="E1:60_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:60_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E4:60_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span></span><a href="binarith.html#K13">*&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="E5:60_1_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:60"><i><font color="Green">E57</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:60_1_2"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E7:60_1_2"/><i><font color="Green">E62</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then </b><i><font color="Green">E63</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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binarith.html#K5">-'</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E5:60_1_2"><i><font color="Green">E60</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1

;<br/><a NAME="E9:60_1_2"/><b>then </b><i><font color="Green">E64</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">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:60"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E7:60_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E10:60_1_2"/><i><font color="Green">E65</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">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:60"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E7:60_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan2c.html#E8:60_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><i><font color="Green">E66</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E4:60_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan2c.html#E9:60_1_2"><i><font color="Green">E64</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">c<sub>3</sub></font>

;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan2c.html#T52" target="_self">Th52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:60_1_2_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_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="E1:60_1_2_3_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E1:60_1_2_3"><i><font color="Green">E67</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:60_1_2_3_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_1_2_3_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E11:60_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:60_1_2_3_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:60_1_2_3_1_1"/><i><font color="Green">E71</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:60_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E7:60_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan2c.html#E10:60_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan3.html#T17">JORDAN3:17</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:60_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E7:60_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan3.html#T32">JORDAN3:32</a></i>
;<br/><a NAME="E6:60_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">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_2_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E2:60_1_2_3_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_2_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E7:60_1_2_3_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_2_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E8:60_1_2_3_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60_1_2_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E9:60_1_2_3_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:60_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E10:60_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><a NAME="E11:60_1_2_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:60_1_2_3_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan2c.html#E8:60_1_2_3_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan2c.html#E9:60_1_2_3_1_1"><i><font color="Green">E74</font></i></a></i>;<br/><b>hence </b><a NAME="E12:60_1_2_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:60_1_2_3_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_1_2_3_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E11:60_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:60_1_2_3_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:60_1_2_3_1_2"/><i><font color="Green">E71</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:60_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E7:60_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan2c.html#E10:60_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan3.html#T17">JORDAN3:17</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:60_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E5:60_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan2c.html#E6:60_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan2c.html#E7:60_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan3.html#T32">JORDAN3:32</a></i>
;<br/><a NAME="E6:60_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">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E11:60_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_2_3_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E2:60_1_2_3_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_2_3_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E7:60_1_2_3_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:60"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan2c.html#E3:60_1_2_3_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><a NAME="E8:60_1_2_3_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:60_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan2c.html#E1:60_1_2_3_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self">Th53</a></i>;<br/><a NAME="E9:60_1_2_3_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:60_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan2c.html#E4:60_1_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E10:60_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><a NAME="E11:60_1_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:60_1_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan2c.html#E8:60_1_2_3_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan2c.html#E9:60_1_2_3_1_2"><i><font color="Green">E74</font></i></a></i>;<br/><b>hence </b><a NAME="E12:60_1_2_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="E14:60_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
