<?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/><b>let </b><font color="Maroon">q</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/><b>let </b><font color="Maroon">r</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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span></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">r</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">q</font></span>)</span></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">r</font></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"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">r</font></span>)</span></span>)</span>

<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> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:48"/><i><font color="Green">E33</font></i>: 
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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><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="E1:48_2"/><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/>

<a NAME="E2:48_2"/>
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font> <a href="ordinal1.html#R1">c=</a>  <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> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, </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/><br/>


</div><b>end;</b></div>
<a NAME="E3:48"/>
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> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</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> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</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="48_3"><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="E1:48_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span>
 ;<br/>

<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_1"/><i><font color="Green">E47</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">q</font>
 ;<br/><a NAME="E2:48_3_1"/><b>then </b>
<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> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <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> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</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> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</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> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T1" target="_self">Th1</a>, </i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_3_2"/>
not <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">q</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:48_3_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">r</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</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> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</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> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T1" target="_self">Th1</a>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</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> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font></span>)</span> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
