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



<i><font color="Green">E35</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E36</font></i>:  <a href="finseq_1.html#K4">dom</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="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<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#K4">^</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<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#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E42</font></i>:  <a href="finseq_1.html#K4">dom</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="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E43</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E5:21"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</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="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_5"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<a NAME="E2:21_5"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_5_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_5_1_1_1"><b>suppose </b></a><a NAME="E1:21_5_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><i><font color="Green">E54</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <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">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</font></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"><font color="Maroon">F</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</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#D7">FINSEQ_1:def 7</a></i>
;<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="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</font> = 
<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">x</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E4:21_5_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <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">x</font> <a href="hidden.html#R1">=</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="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_5_1_1_2"><b>suppose </b></a><a NAME="E1:21_5_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:21_5_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:21_5_1_1_2"/><i><font color="Green">E57</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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <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">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</font></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"><font color="Maroon">G</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</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#D7">FINSEQ_1:def 7</a></i>
;<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="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</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">n</font>
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E7:21_5_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <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">x</font> <a href="hidden.html#R1">=</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="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:21_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:21"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="wsierp_1.html#K4">^</a> <font color="Maroon">G</font></span>)</span> <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">F</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">(<span class="default"><font color="Maroon">G</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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