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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>

<i><font color="Green">E24</font></i>:  <a href="afinsq_1.html#K2" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></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><a class="ref" href="afinsq_1.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0
<b>by </b><i><a class="ref" href="afinsq_1.html#T18" target="_self">Th18</a></i>
<br/>.= 
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></i>
;<br/>
<b>thus </b><a NAME="E2:50"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E2:50_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal4.html#K1">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E1:50"><i><font color="Green">E24</font></i></a>, <a class="ref" href="afinsq_1.html#T10" target="_self">Th10</a></i>;<br/>


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

<i><font color="Green">E25</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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T20" target="_self">Th20</a></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T18" target="_self">Th18</a></i>
<br/>.= 
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="afinsq_1.html#D1" target="_self">Def1</a></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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:50_4_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal4.html#K1">^</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T18" target="_self">Th18</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:50_4_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="afinsq_1.html#D4" target="_self">Def4</a></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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E3:50"><i><font color="Green">E25</font></i></a>, <a class="ref" href="afinsq_1.html#T10" target="_self">Th10</a></i>;<br/>


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


</div>
