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

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



<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">x</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c4">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:56"/><i><font color="Green" title="E49">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="jordan2c.html#T54" target="_self" title="JORDAN2C:th.54">Th54</a></i>;<br/>
<a NAME="E5:56"/><i><font color="Green" title="E50">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E6:56"/><b>then </b><i><font color="Green" title="E51">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1"/>
( <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>suppose </b></a><a NAME="E1:56_1_1"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c5">g</font> = <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><a NAME="E3:56_1_1"/><i><font color="Green" title="E53">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E4:56_1_1"/><i><font color="Green" title="E54">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><i><font color="Green" title="E55">A8</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E6:56"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/><b>then </b><i><font color="Green" title="E56">A9</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> = 
1 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E3:56_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>

;<br/><b>then reconsider </b><font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c5">g</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="jordan2c.html#T52" target="_self" title="JORDAN2C:th.52">Th52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:56_1_1_3"/><i><font color="Green" title="E57">A10</font></i>: 
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_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="E1:56_1_1_3_1"/>
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E2:56_1_1_3"><i><font color="Green" title="E57">A10</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_3_1_1"/><i><font color="Green" title="E58">A11</font></i>: 
<font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font>
 ;<br/><div class="add"><a NAME="E2:56_1_1_3_1_1"/><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:56_1_1_3_1_1"/><i><font color="Green" title="E60">A13</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:56_1_1_3_1_1"/><i><font color="Green" title="E61">A14</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:56_1_1_3_1_1"/><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:56_1_1_3_1_1"/><i><font color="Green" title="E63">A16</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E7:56_1_1_3_1_1"/>
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E8:56_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E64">A17</font></i>: 
<font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><i><font color="Green" title="E65">A18</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c4">i2</font>

;<br/><a NAME="E10:56_1_1_3_1_1"/>
( 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E4:56_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_1_3_1_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_1_3_1_1"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_1_3_1_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="txt" href="jordan2c.html#E8:56_1_1_3_1_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="jordan2c.html#E9:56_1_1_3_1_1"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><a NAME="E12:56_1_1_3_1_1"/><b>then </b>
1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_1_3_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="jordan2c.html#E2:56_1_1_3_1_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_1_3_1_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E13:56_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E66">A19</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_1_3_1_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E14:56_1_1_3_1_1"/><i><font color="Green" title="E67">A20</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56_1_1_3_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_1_3_1_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E15:56_1_1_3_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:56_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_1_3_1_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E16:56_1_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E13:56_1_1_3_1_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="txt" href="jordan2c.html#E14:56_1_1_3_1_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></i>;<br/><b>hence </b><a NAME="E17:56_1_1_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_3_1_2"/><i><font color="Green" title="E58">A21</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font>
 ;<br/><div class="add"><a NAME="E2:56_1_1_3_1_2"/><i><font color="Green" title="E59">A22</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:56_1_1_3_1_2"/><i><font color="Green" title="E60">A23</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:56_1_1_3_1_2"/><i><font color="Green" title="E61">A24</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:56_1_1_3_1_2"/><i><font color="Green" title="E62">A25</font></i>: 
<font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:56_1_1_3_1_2"/><i><font color="Green" title="E63">A26</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E7:56_1_1_3_1_2"/>
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E8:56_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E64">A27</font></i>: 
<font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><i><font color="Green" title="E65">A28</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c4">i2</font>

;<br/><a NAME="E10:56_1_1_3_1_2"/>
( 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan2c.html#E4:56_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_1_3_1_2"><i><font color="Green" title="E61">A24</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_1_3_1_2"><i><font color="Green" title="E62">A25</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_1_3_1_2"><i><font color="Green" title="E63">A26</font></i></a>, <a class="txt" href="jordan2c.html#E8:56_1_1_3_1_2"><i><font color="Green" title="E64">A27</font></i></a>, <a class="txt" href="jordan2c.html#E9:56_1_1_3_1_2"><i><font color="Green" title="E65">A28</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><a NAME="E12:56_1_1_3_1_2"/><b>then </b>
1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_1_3_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="jordan2c.html#E2:56_1_1_3_1_2"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_1_3_1_2"><i><font color="Green" title="E60">A23</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E13:56_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E66">A29</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_1_3_1_2"><i><font color="Green" title="E60">A23</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E14:56_1_1_3_1_2"/><i><font color="Green" title="E67">A30</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:56_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_1_3_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_1_3_1_2"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E15:56_1_1_3_1_2"/>
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:56_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_1_3_1_2"><i><font color="Green" title="E61">A24</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E16:56_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E13:56_1_1_3_1_2"><i><font color="Green" title="E66">A29</font></i></a>, <a class="txt" href="jordan2c.html#E14:56_1_1_3_1_2"><i><font color="Green" title="E67">A30</font></i></a>, <a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></i>;<br/><b>hence </b><a NAME="E17:56_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="E9:56_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><b>suppose </b></a><a NAME="E1:56_1_2"/>
<font color="Maroon" title="c4">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:56_1_2"/><b>then </b><i><font color="Green" title="E52">A31</font></i>: 
<font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">g</font> = <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><a NAME="E4:56_1_2"/><i><font color="Green" title="E53">A32</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E5:56_1_2"/><i><font color="Green" title="E54">A33</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:56_1_2"/><i><font color="Green" title="E55">A34</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E7:56_1_2"/><i><font color="Green" title="E56">A35</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then </b><i><font color="Green" title="E57">A36</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E5:56_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1

;<br/><a NAME="E9:56_1_2"/><b>then </b><i><font color="Green" title="E58">A37</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:56"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="jordan2c.html#E7:56_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E10:56_1_2"/><i><font color="Green" title="E59">A38</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:56"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="jordan2c.html#E7:56_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="jordan2c.html#E8:56_1_2"><i><font color="Green" title="E57">A36</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><i><font color="Green" title="E60">A39</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E4:56_1_2"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="jordan2c.html#E9:56_1_2"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>

;<br/><b>then reconsider </b><font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c5">g</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="jordan2c.html#T52" target="_self" title="JORDAN2C:th.52">Th52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:56_1_2_3"/><i><font color="Green" title="E61">A40</font></i>: 
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_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="E1:56_1_2_3_1"/>
( <font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E2:56_1_2_3"><i><font color="Green" title="E61">A40</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:56_1_2_3_1_1"/><i><font color="Green" title="E62">A41</font></i>: 
<font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font>
 ;<br/><div class="add"><a NAME="E2:56_1_2_3_1_1"/><i><font color="Green" title="E63">A42</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E11:56_1_2"><i><font color="Green" title="E60">A39</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:56_1_2_3_1_1"/><i><font color="Green" title="E64">A43</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:56_1_2_3_1_1"/><i><font color="Green" title="E65">A44</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:56_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="jordan2c.html#E7:56_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="jordan2c.html#E10:56_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="jordan3.html#T17" title="JORDAN3:th.17">JORDAN3:17</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:56_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="jordan2c.html#E7:56_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="jordan3.html#T32" title="JORDAN3:th.32">JORDAN3:32</a></i>
;<br/><a NAME="E6:56_1_2_3_1_1"/><b>then </b>
1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_2_3_1_1"><i><font color="Green" title="E62">A41</font></i></a>, <a class="txt" href="jordan2c.html#E2:56_1_2_3_1_1"><i><font color="Green" title="E63">A42</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_2_3_1_1"><i><font color="Green" title="E64">A43</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E7:56_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E66">A45</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_2_3_1_1"><i><font color="Green" title="E64">A43</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E8:56_1_2_3_1_1"/><i><font color="Green" title="E67">A46</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56_1_2_3_1_1"><i><font color="Green" title="E62">A41</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_2_3_1_1"><i><font color="Green" title="E65">A44</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E9:56_1_2_3_1_1"/><i><font color="Green" title="E68">A47</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:56_1_2"><i><font color="Green" title="E60">A39</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_2_3_1_1"><i><font color="Green" title="E65">A44</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E10:56_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/><b>hence </b><a NAME="E11:56_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:56_1_2_3_1_1"><i><font color="Green" title="E66">A45</font></i></a>, <a class="txt" href="jordan2c.html#E8:56_1_2_3_1_1"><i><font color="Green" title="E67">A46</font></i></a>, <a class="txt" href="jordan2c.html#E9:56_1_2_3_1_1"><i><font color="Green" title="E68">A47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:56_1_2_3_1_2"/><i><font color="Green" title="E62">A48</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">y2</font>
 ;<br/><div class="add"><a NAME="E2:56_1_2_3_1_2"/><i><font color="Green" title="E63">A49</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E11:56_1_2"><i><font color="Green" title="E60">A39</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:56_1_2_3_1_2"/><i><font color="Green" title="E64">A50</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E4:56_1_2_3_1_2"/><i><font color="Green" title="E65">A51</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:56_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="jordan2c.html#E7:56_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="jordan2c.html#E10:56_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="jordan3.html#T17" title="JORDAN3:th.17">JORDAN3:17</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E2:56_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="jordan2c.html#E5:56_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="jordan2c.html#E6:56_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="jordan2c.html#E7:56_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="ref" href="jordan3.html#T32" title="JORDAN3:th.32">JORDAN3:32</a></i>
;<br/><a NAME="E6:56_1_2_3_1_2"/><b>then </b>
1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E5:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="jordan2c.html#E11:56_1_2"><i><font color="Green" title="E60">A39</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_2_3_1_2"><i><font color="Green" title="E62">A48</font></i></a>, <a class="txt" href="jordan2c.html#E2:56_1_2_3_1_2"><i><font color="Green" title="E63">A49</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_2_3_1_2"><i><font color="Green" title="E64">A50</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E7:56_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E66">A52</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="jordan2c.html#E3:56_1_2_3_1_2"><i><font color="Green" title="E64">A50</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E8:56_1_2_3_1_2"/><i><font color="Green" title="E67">A53</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:56_1_2"><i><font color="Green" title="E60">A39</font></i></a>, <a class="txt" href="jordan2c.html#E1:56_1_2_3_1_2"><i><font color="Green" title="E62">A48</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_2_3_1_2"><i><font color="Green" title="E65">A51</font></i></a>, <a class="ref" href="jordan2c.html#T53" target="_self" title="JORDAN2C:th.53">Th53</a></i>;<br/><a NAME="E9:56_1_2_3_1_2"/><i><font color="Green" title="E68">A54</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:56_1_2"><i><font color="Green" title="E60">A39</font></i></a>, <a class="txt" href="jordan2c.html#E4:56_1_2_3_1_2"><i><font color="Green" title="E65">A51</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
<font color="Maroon" title="c5">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1
<b>by </b><i><a class="txt" href="jordan2c.html#E10:56_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/><b>hence </b><a NAME="E11:56_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:56_1_2_3_1_2"><i><font color="Green" title="E66">A52</font></i></a>, <a class="txt" href="jordan2c.html#E8:56_1_2_3_1_2"><i><font color="Green" title="E67">A53</font></i></a>, <a class="txt" href="jordan2c.html#E9:56_1_2_3_1_2"><i><font color="Green" title="E68">A54</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:56_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> st <br/>for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( not <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c2">x</font> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">y</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
