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

<b>let </b><font color="Maroon">p</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>

<i><font color="Green">E31</font></i>:  <a href="afinsq_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> = 
 <a href="afinsq_1.html#K1" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
<b>by </b><i/>
<br/>.= 
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/>
<b>thus </b><a NAME="E2:50"/>
<font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<a NAME="E1:50_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:50_2"/>
<font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<i><font color="Green">E33</font></i>:  <a href="afinsq_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="afinsq_1.html#K1" target="_self">len</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font>
<b>by </b><i/>
;<br/>
<b>thus </b><a NAME="E4:50"/>
<a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>proof </b></a><div class="add">

<a NAME="E1:50_4"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:50_4_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_4"/>
<a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>


</div>
