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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="rearran1.html#NM1">RearrangmentGen</a> of <font color="Maroon">D</font>;<br/>



<b>set </b><font color="Maroon">C</font> =  <a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span>;<br/>
<a NAME="E1:22"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <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">A</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p3">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:22"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:22"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <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">A</font></span>)</span> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <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">A</font></span>)</span>
 ;<br/>

<a NAME="E2:22_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:22_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>case </b></a><a NAME="E1:22_1_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:22_1_1_1_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><b>case </b></a><a NAME="E1:22_1_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:22_1_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:22_1_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:22_1_1_1_2"/><i><font color="Green">E49</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:22_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:22_1_1_1_2"/>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/><a NAME="E9:22_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>

;<br/><b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">D</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T48">XBOOLE_1:48</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
 <a href="rearran1.html#K2" target="_self">Co_Gen</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" target="_self">Co_Gen</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div>
