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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</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">p</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</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="E2:43"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<a NAME="E4:43"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E5:43"/><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="xxreal_0.html#NR3" target="_self">&lt;</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="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:43"/><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> <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="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:43"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <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/>


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