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<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>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>assume </b><a NAME="E1:41"/>
<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>
 ;<br/>

<a NAME="E2:41"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <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> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:41"/><b>then </b>
<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/>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E31</font></i>: 
<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="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1"/>
not  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:41_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E3:41_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">p</font> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">q</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">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_2"/>
 <a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green">E43</font></i>: 
<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">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:41_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" target="_self">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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="xxreal_0.html#NR3" target="_self">&lt;</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">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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>, , <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E5:41_2"/><b>then </b>
<font color="Maroon">n</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><a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E6:41_2"/><b>then </b>
<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">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:41_2"/>
( <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> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">q</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">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:41"/>
( <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> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">q</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">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> ) )
 <b>by </b><i/>;<br/>


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