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

<b>let </b><font color="Maroon" title="c1">fp</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E37">A0</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; ( for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">fp2</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c3">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:53"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">fp</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">fp2</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">q</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp2</font>;<br/>
<a NAME="E4:53"/><i><font color="Green" title="E39">B0</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</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> <font color="Maroon" title="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E3:53"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">l</font> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,$1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>($1);<br/>
<b>let </b><font color="Maroon" title="c5">fp1</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E5:53"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">fp1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font>
 ;<br/>

<a NAME="E6:53"/><i><font color="Green" title="E40">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:53_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53"><i><font color="Green" title="E37">A0</font></i></a>, <a class="ref" href="int_4.html#T31" target="_self" title="INT_4:th.31">Th28</a></i>;<br/>
<a NAME="E4:53_1"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E40"><i><font color="Green" title="E29">Lm6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">l</font> = <font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c6">a</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E6:53_1"/><i><font color="Green" title="E43">A8</font></i>: 
 <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c7">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E4:53_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<a NAME="E7:53_1"/>
<font color="Maroon" title="c7">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53"><i><font color="Green" title="E37">A0</font></i></a>, <a class="txt" href="int_4.html#E2:53_1"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E8:53_1"/><b>then </b>
<font color="Maroon" title="c7">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:53_1"/><i><font color="Green" title="E44">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="int_4.html#E6:53_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="int_4.html#T16" target="_self" title="INT_4:th.16">Th13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c8">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">n</font>
;<br/>

<b>thus </b><a NAME="E12:53_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c9">n</font>]
 <b>by </b><i><a class="txt" href="int_4.html#E10:53_1"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">s</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:53"/><i><font color="Green" title="E41">A6</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span> &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">a</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="int_4.html#E6:53"><i><font color="Green" title="E40">A2</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,$1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> $1</span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c7">s1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E10:53"/><i><font color="Green" title="E42">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">s1</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> <font color="Maroon" title="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E11:53"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s1</font> holds <br/><font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">s1</font>
 ;<br/>

<a NAME="E2:53_2"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">s1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:53_2"/><b>then </b>
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:53_2"/>
<font color="Maroon" title="c7">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">s1</font> = <font color="Maroon" title="c7">s1</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T14" title="FINSEQ_2:th.14">FINSEQ_2:14</a></i>;<br/>
<a NAME="E13:53"/><i><font color="Green" title="E43">A10</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:53_3"/><i><font color="Green" title="E44">A11</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b</font> )
 ;<br/>

<a NAME="E2:53_3"/><b>then </b><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E4:53"><i><font color="Green" title="E39">B0</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:53_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53"><i><font color="Green" title="E37">A0</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:53_3"/><b>then </b>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c9">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53_3"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="int_4.html#T35" target="_self" title="INT_4:th.35">Th32</a></i>;<br/>
<a NAME="E5:53_3"/><b>then </b>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<a NAME="E6:53_3"/><b>then </b>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E7:53_3"/>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="int_4.html#E2:53_3"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="int_4.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:53"/><i><font color="Green" title="E44">A25</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Olive" title="b1">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53_4"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font>
 ;<br/>

<a NAME="E2:53_4"/><b>then </b>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="int_4.html#E4:53"><i><font color="Green" title="E39">B0</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:53_4"/><b>then </b><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:53_4"/><b>then </b><i><font color="Green" title="E47">A15</font></i>: 
<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="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></i>;<br/>
<a NAME="E5:53_4"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span> holds <br/><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53_4_1"/><i><font color="Green" title="E48">A16</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span>
 ;<br/>

<a NAME="E2:53_4_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T47" title="WSIERP_1:th.47">WSIERP_1:47</a></i>;<br/>
<a NAME="E3:53_4_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53_4_1"><i><font color="Green" title="E48">A16</font></i></a></i>;<br/>
<a NAME="E4:53_4_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:53_4_1"/><b>then </b><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E4:53"><i><font color="Green" title="E39">B0</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:53_4_1"/><i><font color="Green" title="E50">A18</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">a</font> <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="c2">fp2</font> &amp; <font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c9">b</font> <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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53_4"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="int_4.html#E4:53_4"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="int_4.html#E1:53_4_1"><i><font color="Green" title="E48">A16</font></i></a>, <a class="txt" href="int_4.html#E4:53"><i><font color="Green" title="E39">B0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_4_1_1"/>
( <font color="Maroon" title="c10">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">b</font> or <font color="Maroon" title="c10">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_1"><b>suppose </b></a><a NAME="E1:53_4_1_1_1"/><i><font color="Green" title="E51">A19</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">b</font>
 ;<br/><div class="add"><a NAME="E2:53_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="int_4.html#E3:53_4"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:53_4_1_1_1"/>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="int_4.html#E1:53_4"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="int_4.html#E5:53_4_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="int_4.html#E1:53_4_1_1_1"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="int_4.html#E13:53"><i><font color="Green" title="E43">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_2"><b>suppose </b></a><a NAME="E1:53_4_1_1_2"/><i><font color="Green" title="E51">A21</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">b</font>
 ;<br/><div class="add"><a NAME="E2:53_4_1_1_2"/><b>then </b><i><font color="Green" title="E52">A22</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E3:53_4"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></i>;<br/><a NAME="E3:53_4_1_1_2"/><i><font color="Green" title="E53">A23</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E6:53_4_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:53_4_1_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E6:53_4_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E5:53_4_1_1_2"/><b>then </b><i><font color="Green" title="E54">A24</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font>
 <b>by </b><i><a class="txt" href="int_4.html#E4:53"><i><font color="Green" title="E39">B0</font></i></a>, <a class="txt" href="int_4.html#E3:53_4_1_1_2"><i><font color="Green" title="E53">A23</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:53_4_1_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="int_4.html#E1:53_4_1_1_2"><i><font color="Green" title="E51">A21</font></i></a>, <a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a></i>;<br/><b>hence </b><a NAME="E7:53_4_1_1_2"/>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="int_4.html#E1:53_4"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="int_4.html#E5:53_4_1_1_2"><i><font color="Green" title="E54">A24</font></i></a>, <a class="txt" href="int_4.html#E13:53"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="int_4.html#E2:53_4_1_1_2"><i><font color="Green" title="E52">A22</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="E6:53_4"/>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_4.html#T36" target="_self" title="INT_4:th.36">Th33</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c9">x</font> =  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c8">s1</font>;<br/>
<a NAME="E15:53"/><i><font color="Green" title="E45">A29</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:53_5"/><i><font color="Green" title="E46">A26</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font>
 ;<br/>

<a NAME="E2:53_5"/><b>then </b><i><font color="Green" title="E47">A27</font></i>: 
( <font color="Maroon" title="c10">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">b</font> <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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="int_4.html#E4:53"><i><font color="Green" title="E39">B0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:53_5"/>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53_5"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="int_4.html#E14:53"><i><font color="Green" title="E44">A25</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">w</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:53_5"/><i><font color="Green" title="E48">A34</font></i>: 
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">w</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E6:53_5"/>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="int_4.html#E1:53"><i><font color="Green" title="E37">A0</font></i></a>, <a class="txt" href="int_4.html#E1:53_5"><i><font color="Green" title="E46">A26</font></i></a></i>;<br/>
<a NAME="E7:53_5"/><b>then </b><i><font color="Green" title="E49">A28</font></i>: 
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E8:53_5"/><i><font color="Green" title="E50">A29</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c2">fp2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E2:53_5"><i><font color="Green" title="E47">A27</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E51">A30</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span>

;<br/>
<a NAME="E10:53_5"/>
<font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">b</font>)
 
<b>by </b><i><a class="txt" href="int_4.html#E8:53"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="int_4.html#E8:53_5"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="int_4.html#E4:53"><i><font color="Green" title="E39">B0</font></i></a></i>;<br/>
<a NAME="E11:53_5"/><b>then </b><i><font color="Green" title="E52">A31</font></i>: 
 ex <font color="Olive" title="b1">ll</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">ll</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">ll</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
;<br/>
<a NAME="E12:53_5"/><i><font color="Green" title="E53">A36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c12">l</font> =  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font></span>)</span> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E2:53_5_2"/>
<font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c12">l</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E5:53_5"><i><font color="Green" title="E48">A34</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E3:53_5_2"/><b>then </b><i><font color="Green" title="E54">C1</font></i>: 
<font color="Maroon" title="c12">l</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="int_4.html#E7:53_5"><i><font color="Green" title="E49">A28</font></i></a>, <a class="ref" href="int_1.html#T89" title="INT_1:th.89">INT_1:89</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="int_3.html#T14" title="INT_3:th.14">INT_3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E3:53_5_2"><i><font color="Green" title="E54">C1</font></i></a>, <a class="ref" href="int_3.html#T13" title="INT_3:th.13">INT_3:13</a></i>
;<br/>
<b>hence </b><a NAME="E5:53_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:53_5"/>
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="int_4.html#E2:53_5"><i><font color="Green" title="E47">A27</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8" title="WSIERP_1:func.8">Del</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">s1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T27" title="WSIERP_1:th.27">WSIERP_1:27</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="int_4.html#E11:53_5"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="int_4.html#E7:53_5"><i><font color="Green" title="E49">A28</font></i></a>, <a class="txt" href="int_4.html#E9:53_5"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="int_4.html#E48"><i><font color="Green" title="E35">Lm7</font></i></a>, <a class="txt" href="int_4.html#E12:53_5"><i><font color="Green" title="E53">A36</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E15:53_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c8">s1</font>
;<br/>

<b>thus </b><a NAME="E16:53"/>
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">fp1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="int_4.html#E15:53"><i><font color="Green" title="E45">A29</font></i></a></i>;<br/>


</div>
