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<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E1:58"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">r</font> or <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font> )
 ;<br/>

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">p</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:58_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" 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="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:58_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" 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="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:58_1"/><b>then </b>
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E35</font></i>:  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font> = 
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
<b>by </b><i/>
;<br/><a NAME="E6:58_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="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_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="E1:58_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>hence </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7" target="_self">^</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="txt" href="finseq_1.html#E5:58_1"><i><font color="Green">E35</font></i></a>, </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="txt" href="finseq_1.html#E1:58_1_2"><i><font color="Green">E36</font></i></a>, , </i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:58_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font>
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="finseq_1.html#K3" target="_self">len</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E3:58_2"/><b>then </b>
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E51</font></i>:  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font> = 
 <a href="finseq_1.html#K2" target="_self">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">q</font>
<b>by </b><i/>
;<br/><a NAME="E5:58_2"/>
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="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_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="E1:58_2_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4" target="_self">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>hence </b><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7" target="_self">^</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="finseq_1.html#K3" target="_self">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </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="txt" href="finseq_1.html#E1:58_2"><i><font color="Green">E37</font></i></a>, </i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:58_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="finseq_1.html#E2:58"><i><font color="Green">E33</font></i></a>, </i>;<br/>


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