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

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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">f</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Olive">f</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>;<br/>
<a NAME="E1:17"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E3:17_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:17_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:17"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">f</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:17_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>reconsider </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E4:17_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:17_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:17_2"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">fn</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>
;<br/>
<a NAME="E8:17_2"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:17_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:17_2_2"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:17_2_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:17_2_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_2_2_1"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/>

<a NAME="E2:17_2_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:17_2_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E4:17_2_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:17_2_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:17_2_2_1"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:17_2_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fn</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E9:17_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:17_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:17_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E12:17_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:17_2"/><b>then </b>
<font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p3">(<span class="default"><font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</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#K8">Product</a> <span class="p4">(<span class="default"><font color="Maroon">fn</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><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#K8">Product</a> <font color="Maroon">fn</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, , </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#K8">Product</a> <font color="Maroon">fn</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E15:17_2"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:17"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">f</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:17"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E5:17"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E6:17"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/>


</div>
