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<div class="add">

<b>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">z</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">m</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> ;<br/>





<a NAME="E1:16"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E2:16"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:16"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">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">i</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">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</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">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="finseq_2.html#T8">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">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#T2">BINARITH:2</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T2">BINARITH:2</a></i>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D4">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">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><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">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <font color="Maroon">d</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T3">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">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T3">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">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="binarith.html#K10">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K4">'not'</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K10">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">d</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, , </i>;<br/>


</div>
