<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:16"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="idea_1.html#K1" target="_self">ZERO</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E2:16"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K1" target="_self">ZERO</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><i><font color="Green">E39</font></i>: <span class="p1">(<span class="default">0 <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2 = 
0 <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="nat_2.html#T4">NAT_2:4</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>
;<br/><a NAME="E3:16_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="idea_1.html#K1" target="_self">ZERO</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><b>then </b><i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><a href="idea_1.html#K1" target="_self">ZERO</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binari_3.html#K1">-BinarySequence</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> 2</span>)</span>,0,<a href="margrel1.html#K7">FALSE</a> ,<a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i>, <a class="ref" href="binari_3.html#D1">BINARI_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K1" target="_self">ZERO</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="idea_1.html#K1" target="_self">ZERO</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
<font color="Maroon">n</font> <a href="binari_3.html#K1">-BinarySequence</a> 0 <a href="hidden.html#R1">=</a>  <a href="idea_1.html#K1" target="_self">ZERO</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
