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

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:82"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>

<a NAME="E2:82"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:82"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T11" target="_self">Th11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">b</font> = 1;<br/><b>take </b><font color="Maroon">e</font> = 1;<br/><b>thus </b><a NAME="E2:82_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><b>thus </b><a NAME="E3:82_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>;<br/><b>thus </b><a NAME="E4:82_1_1"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font> or <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="int_2.html#T16">INT_2:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:82_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E19"><i><font color="Green">Lemma58</font></i></a></i>;<br/><b>set </b><font color="Maroon">d</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span>;<br/><b>set </b><font color="Maroon">d1</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1;<br/><a NAME="E3:82_1_2"/><i><font color="Green">E105</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/><a NAME="E4:82_1_2"/><i><font color="Green">E106</font></i>: 
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 )
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/><a NAME="E5:82_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T15" target="_self">Th15</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:82_1_2"/>
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="square_1.html#T83">SQUARE_1:83</a>, <a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/><a NAME="E7:82_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E8:82_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="wsierp_1.html#T16" target="_self">Th16</a></i>;<br/><b>then reconsider </b><font color="Maroon">d</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E10:82_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">d</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i/>;<br/><a NAME="E11:82_1_2"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">d1</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a></i>;<br/><b>set </b><font color="Maroon">e1</font> = <font color="Maroon">d1</font> <a href="square_1.html#K5">^2</a> ;<br/><a NAME="E13:82_1_2"/>
<font color="Maroon">d1</font> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">d1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">d1</font> <a href="square_1.html#K5">^2</a>  as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E15:82_1_2"/>
 <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T16" target="_self">Th16</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E16:82_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font>
 <b>by </b><i>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/><a NAME="E17:82_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>;<br/><b>consider </b><font color="Maroon">fs</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E19:82_1_2"/><i><font color="Green">E113</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; ( for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">e1</font> holds <br/><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E20:82_1_2"/><i><font color="Green">E114</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E21:82_1_2"/><i><font color="Green">E115</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">fs</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:82_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fs</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:82_1_2_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> &amp; <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:82_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">e1</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="wsierp_1.html#E36"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:82_1_2_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="wsierp_1.html#E36"><i><font color="Green">Lemma74</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E7:82_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<a NAME="E8:82_1_2_1"/><b>then </b><i><font color="Green">E124</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E38"><i><font color="Green">Lemma75</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E10:82_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/>
<a NAME="E11:82_1_2_1"/><b>then </b>
<font color="Maroon">v1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E38"><i><font color="Green">Lemma75</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E12:82_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E39"><i><font color="Green">Lemma76</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:82_1_2"/>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">e1</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E23:82_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">fs</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="wsierp_1.html#E35"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E24:82_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">fs</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a>, <a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="wsierp_1.html#E35"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E25:82_1_2"/><b>then </b>
not <font color="Maroon">fs</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E36"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="finseq_4.html#T74">FINSEQ_4:74</a></i>;<br/><b>then consider </b><font color="Maroon">v1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E27:82_1_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> &amp; <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v1</font>, <font color="Maroon">v2</font> = <font color="Maroon">v2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="wsierp_1.html#E38"><i><font color="Green">Lemma75</font></i></a></i>;<br/><a NAME="E29:82_1_2"/><i><font color="Green">E127</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">v1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="wsierp_1.html#E38"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon">x1</font> = <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font>;<br/><b>set </b><font color="Maroon">x2</font> = <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font>;<br/><b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span>;<br/><b>set </b><font color="Maroon">y1</font> = <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font>;<br/><b>set </b><font color="Maroon">y2</font> = <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font>;<br/><b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span>;<br/><a NAME="E30:82_1_2"/><i><font color="Green">E143</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1_2_2"/>
( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:82_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font> )
 
;<br/>
<b>then </b><font color="Maroon">v1</font> <a href="real_1.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">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="newton.html#T80">NEWTON:80</a></i>
<br/>.= 
<font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="newton.html#T80">NEWTON:80</a></i>
;<br/>
<b>hence </b><a NAME="E4:82_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="wsierp_1.html#E38"><i><font color="Green">Lemma75</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E31:82_1_2"/><i><font color="Green">E144</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/><a NAME="E32:82_1_2"/><i><font color="Green">E145</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/><a NAME="E33:82_1_2"/><i><font color="Green">E146</font></i>: 
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:82_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:82_1_2_3"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:82_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:82_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:82_1_2_3"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E6:82_1_2_3"/>
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E40"><i><font color="Green">Lemma81</font></i></a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E34:82_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E35:82_1_2"/><i><font color="Green">E156</font></i>: 
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:82_1_2_4"/><i><font color="Green">E158</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> )
 
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>;<br/>
<a NAME="E2:82_1_2_4"/><b>then </b><i><font color="Green">E159</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> )
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E3:82_1_2_4"/>
( <font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> <a href="real_1.html#K5">-</a> 1 )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E39"><i><font color="Green">Lemma76</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:82_1_2_4"/><b>then </b><i><font color="Green">E160</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> )
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E5:82_1_2_4"/><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="xcmplx_1.html#T128">XCMPLX_1:128</a></i>;<br/>
<a NAME="E6:82_1_2_4"/>
1 <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E7:82_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:82_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:82_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d1</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:82_1_2_4"/><b>then </b><i><font color="Green">E162</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E11:82_1_2_4"/><i><font color="Green">E163</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default">0</span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>;<br/>
<a NAME="E12:82_1_2_4"/>
( <font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E39"><i><font color="Green">Lemma76</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E13:82_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/>
<a NAME="E14:82_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="pre_ff.html#T11">PRE_FF:11</a></i>;<br/>
<a NAME="E15:82_1_2_4"/><b>then </b><i><font color="Green">E164</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:82_1_2_4"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E17:82_1_2_4"/>
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:82_1_2"/><b>then </b><i><font color="Green">E165</font></i>: 
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E37:82_1_2"/>
( <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a>, <a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="wsierp_1.html#E38"><i><font color="Green">Lemma75</font></i></a></i>;<br/><a NAME="E38:82_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E38"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="xcmplx_1.html#T2">XCMPLX_1:2</a></i>;<br/><a NAME="E39:82_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E40:82_1_2"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T11" target="_self">Th11</a>, <a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a></i>;<br/><a NAME="E41:82_1_2"/><i><font color="Green">E167</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span>
 ;<br/><b>set </b><font color="Maroon">d2</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span>;<br/><a NAME="E42:82_1_2"/><i><font color="Green">E168</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span></span>)</span>
 ;<br/><div><i><font color="Green">E169</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:82_1_2_5"/><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:82_1_2_5"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a>  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a></i>;<br/><a NAME="E3:82_1_2_5"/><b>then </b>
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E40"><i><font color="Green">Lemma81</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:82_1_2_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E184</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:82_1_2_6"/><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:82_1_2_6"/><b>then </b>
<font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T11" target="_self">Th11</a>, , , <a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/><a NAME="E3:82_1_2_6"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a>  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a></i>;<br/><a NAME="E4:82_1_2_6"/><b>then </b>
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:82_1_2_6"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">b</font> =  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span>;<br/><b>take </b><font color="Maroon">e</font> =  <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span>;<br/><b>thus </b><a NAME="E45:82_1_2"/>
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/><b>thus </b><a NAME="E46:82_1_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T16" target="_self">Th16</a>, <a class="txt" href="wsierp_1.html#E41"><i><font color="Green">Lemma82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E47:82_1_2"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T16" target="_self">Th16</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E48:82_1_2"/>
( ( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K5">div</a> <font color="Maroon">d1</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> or <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">d1</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="absvalue.html#T1">ABSVALUE:1</a></i>;<br/><b>hence </b><a NAME="E49:82_1_2"/>
( <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">e</font> or <font color="Maroon">a</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">e</font> )
 <b>by </b><i>, , <a class="ref" href="int_2.html#T14">INT_2:14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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