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<b>let </b><font color="Maroon">C</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">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">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="rearran1.html#NM1">RearrangmentGen</a> of <font color="Maroon">C</font>;<br/>









<b>set </b><font color="Maroon">fd</font> =  <a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">mf</font> =  <a href="rfinseq.html#K2">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>
<b>assume </b><a NAME="E1:27"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">D</font> )
 ;<br/>

<a NAME="E2:27"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c</font> <a href="rfunct_3.html#R1">is_common_for_dom</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T35">RFUNCT_3:35</a></i>;<br/>
<a NAME="E3:27"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18">#</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T36">RFUNCT_3:36</a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#D3">RFINSEQ:def 3</a>, <a class="ref" href="rfunct_3.html#D6">RFUNCT_3:def 6</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E6:27"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> 1 implies <span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> 1
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T29">RFINSEQ:29</a></i>
;<br/><br/>


</div><b>end;</b></div>

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

<b>set </b><font color="Maroon">mfn</font> =  <a href="rfinseq.html#K2">MIM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>assume </b><a NAME="E7:27"/><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/>

<a NAME="E8:27"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16">CHI</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18">#</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K3" target="_self">Rland</a> <font color="Maroon">F</font>,<font color="Maroon">B</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K2">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T111">RVSUM_1:111</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T30">RFINSEQ:30</a></i>
;<br/><br/>


</div>
