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



<i><font color="Green">E32</font></i>: <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_2"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1"><b>suppose </b></a><a NAME="E1:24_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:24_2_1"/><b>then </b>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:24_2_1"/>
(  <a href="jordan4.html#K1" target="_self">S_Drop</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">f</font> &amp;  <a href="jordan4.html#K1" target="_self">S_Drop</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="jordan4.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_2"><b>suppose </b></a><a NAME="E1:24_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:24_2_2"/><b>then </b>
 <a href="jordan4.html#K1" target="_self">S_Drop</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:24_2_2"/>
(  <a href="jordan4.html#K1" target="_self">S_Drop</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_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">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">f</font> &amp;  <a href="jordan4.html#K1" target="_self">S_Drop</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan4.html#K1" target="_self">S_Drop</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="jordan4.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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