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

<b>let </b><font color="Maroon" title="c1">m</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">z</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">m</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon" title="c3">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>





<a NAME="E1:16"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E2:16"/><i><font color="Green" title="E11">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:16"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="binari_2.html#E1:16_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="finseq_2.html#T8" title="FINSEQ_2:th.8">FINSEQ_2:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font>
 ;<br/><div class="add"><i><font color="Green" title="E14">A5</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font> = 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_2.html#E1:16_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="binarith.html#D4" title="BINARITH:def.4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_2.html#E1:16_1_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="binarith.html#T2" title="BINARITH:th.2">BINARITH:2</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="binari_2.html#E1:16_1_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="binarith.html#T2" title="BINARITH:th.2">BINARITH:2</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_2.html#E1:16_1_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="binarith.html#D4" title="BINARITH:def.4">BINARITH:def 4</a></i>
;<br/><b>hence </b><a NAME="E4:16_1_1_1"/>
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="binari_2.html#E2:16_1_1_1"><i><font color="Green" title="E14">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_2"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font> = 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_2.html#E1:16_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="binarith.html#D4" title="BINARITH:def.4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="txt" href="binari_2.html#E1:16_1_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="binarith.html#T3" title="BINARITH:th.3">BINARITH:3</a></i>
;<br/><b>hence </b><a NAME="E3:16_1_1_2"/>
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="binari_2.html#E1:16_1_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="binarith.html#T3" title="BINARITH:th.3">BINARITH:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
 <a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <font color="Maroon" title="c3">d</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="binari_2.html#E1:16"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="binari_2.html#E2:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="binari_2.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>


</div>
