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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:14"/><i><font color="Green">E8</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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="E2:14"/><i><font color="Green">E9</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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="E3:14"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="binari_2.html#E1:14_1"><i><font color="Green">E10</font></i></a>, <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="14_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:14_1_1_1"/>
<span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="binari_2.html#E1:14_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binarith.html#T2">BINARITH:2</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_2.html#E1:14_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="binari_2.html#E1:14_1_1_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="binari_2.html#T7" target="_self">Th7</a></i>
;<br/><b>hence </b><a NAME="E3:14_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="binari_2.html#E1:14_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="binari_2.html#E1:14_1_1_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="binari_2.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="binari_2.html#E1:14_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binarith.html#T2">BINARITH:2</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_2.html#E1:14_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="binari_2.html#E1:14_1_1_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="binari_2.html#T8" target="_self">Th8</a></i>
;<br/><b>hence </b><a NAME="E3:14_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="binari_2.html#E1:14_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="binari_2.html#E1:14_1_1_1_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="binari_2.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:14_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:14_1_1_2"/><b>then </b>
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:14_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="binari_2.html#E1:14_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="binari_2.html#E1:14_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binari_2.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E5:14_1_1_2"/>
<span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="binari_2.html#E1:14_1_1_2"><i><font color="Green">E11</font></i></a>, <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="E4:14"/>
 <a href="binari_2.html#K2" target="_self">Bin1</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" target="_self">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K12">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="binari_2.html#E1:14"><i><font color="Green">E8</font></i></a>, <a class="txt" href="binari_2.html#E2:14"><i><font color="Green">E9</font></i></a>, <a class="txt" href="binari_2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>


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