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<b>let </b><font color="Maroon">pq</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/>

<a NAME="E1:36_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font> &amp;  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">q</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:36_1_1"/><i><font color="Green">E33</font></i>: 
<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="ordinal2.html#K10">+^</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="hidden.html#R1">=</a> <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> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T49">CARD_2:49</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:36_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">pq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font>
 ;<br/><b>hence </b><a NAME="E2:36_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">pq</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">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> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="ordinal4.html#D1">ORDINAL4:def 1</a></i>;<br/><b>thus </b><a NAME="E3:36_1_1_1"/>
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">pq</font> <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>
 <b>by </b><i>, <a class="ref" href="ordinal4.html#D1">ORDINAL4:def 1</a></i>;<br/><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="E4:36_1_1_1"/>
<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="E5:36_1_1_1"/><b>then </b>
<font color="Maroon">pq</font> <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="ordinal2.html#K10">+^</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <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="ordinal4.html#D1">ORDINAL4:def 1</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1"/>
<font color="Maroon">pq</font> <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> <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="card_2.html#T49">CARD_2:49</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E4:36_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">pq</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">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> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E5:36_1_1"/><i><font color="Green">E46</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/><font color="Maroon">pq</font> <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>
 <b>and </b><br/><a NAME="E6:36_1_1"/><i><font color="Green">E47</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">q</font> holds <br/><font color="Maroon">pq</font> <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> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 ;<br/>

<a NAME="E7:36_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">a</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">a</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">pq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:36_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</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 reconsider </b><font color="Maroon">k</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>thus </b><font color="Maroon">pq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">p</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a</font></span>)</span> = 
<font color="Maroon">pq</font> <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>, <a class="ref" href="card_2.html#T49">CARD_2:49</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:36_1_1"/>
<font color="Maroon">pq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , , <a class="ref" href="ordinal4.html#D1">ORDINAL4:def 1</a></i>;<br/>


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