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<div class="add">

<b>let </b><font color="Maroon">F2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">X</font>, <font color="Maroon">F3</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_2"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2</font> holds <br/><font color="Maroon">F2</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K3">`</a>  ) )
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green">E53</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F3</font> holds <br/><font color="Maroon">F3</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K3">`</a>  ) )
 ;<br/>

<a NAME="E3:47_1_2"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F3</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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="relset_1.html#K4">dom</a> <font color="Maroon">F2</font> holds <br/><font color="Maroon">F2</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F3</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Olive">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><i><font color="Green">E58</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F3</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , </i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F3</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:47_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F2</font>
 ;<br/>

<b>hence </b><font color="Maroon">F2</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F3</font> <a href="prob_3.html#K7" target="_self">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_1_2"/>
<font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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