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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="integra1.html#V1" target="_self">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> =  <a href="integra1.html#K3" target="_self">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:46"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( 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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E3:46"/><i><font color="Green">E34</font></i>: 
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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:46_1"/>
 <a href="integra1.html#K3" target="_self">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:46_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E2:46"><i><font color="Green">E33</font></i></a>, <a class="txt" href="integra1.html#E1:46_1"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:46"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<a NAME="E1:46_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:46_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:46_2"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>thus </b><a NAME="E5:46_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="integra1.html#E4:46_2"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:46"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( 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> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:46"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( 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> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E10:46"/><i><font color="Green">E37</font></i>: 
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> <font color="Maroon">c<sub>5</sub></font> holds <br/> <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:46_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E3:46_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:46_3"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E5:46_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_3"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E6:46_3"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:46_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:46_3_1"/><b>then </b>
( 1 <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E3:46_3_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:46_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:46_3"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_3_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</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="46_3_2_1_1"><b>suppose </b></a><a NAME="E1:46_3_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:46_3_2_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E6:46_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E3:46_3_2_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E7:46_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra1.html#E8:46_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra1.html#E1:46_3_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_2_1_2"><b>suppose </b></a><a NAME="E1:46_3_2_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:46_3_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:46_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:46_3_2_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E5:46_3_2_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E6:46_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="integra1.html#E1:46_3_2_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E6:46_3_2_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E7:46_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra1.html#E8:46_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra1.html#E4:46_3_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:46_3"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:46"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4"><b>proof </b></a><div class="add">

<a NAME="E1:46_4"/>
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> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_4_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:46_4_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E3:46_4_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E10:46"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra1.html#E1:46_4_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:46_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra1.html#E1:46_4_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:46"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( 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#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</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"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5"><b>proof </b></a><div class="add">

<a NAME="E1:46_5"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E2:46_5"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:46_5"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:46_5"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:46_5"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra1.html#E2:46_5"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E6:46_5"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra1.html#E4:46_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>

<b>thus </b><a NAME="E7:46_5"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:46"><i><font color="Green">E33</font></i></a>, <a class="txt" href="integra1.html#E5:46_5"><i><font color="Green">E42</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E8:46_5"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E9:46_5"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E8:46_5"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:46_5_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E6:46_5"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:46_5"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:46_5_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or  not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i><a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:46_5_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46_5"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E8:46_5"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E12:46_5"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:46_5"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_5_3_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 </span><b>by </b><i><a class="txt" href="integra1.html#E10:46_5"><i><font color="Green">E46</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3_1_1"><b>suppose </b></a><a NAME="E1:46_5_3_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:46_5_3_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E5:46_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra1.html#E8:46_5"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:46_5_3_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:46_5_3_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:46_5_3_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E6:46_5_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E7:46_5_3_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E11:46_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E8:46_5_3_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E9:46_5_3_1_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E10:46_5_3_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E12:46_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E11:46_5_3_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="integra1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E12:46_5_3_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E3:46_5_3_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E4:46_5_3_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E13:46_5_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:46"><i><font color="Green">E34</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><b>hence </b><a NAME="E14:46_5_3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:46_5_3_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E4:46_5_3_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E7:46_5_3_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E10:46_5_3_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra1.html#E12:46_5_3_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3_1_2"><b>suppose </b></a><a NAME="E1:46_5_3_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><a NAME="E2:46_5_3_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_1"/>
1 <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K10">-</a> 1
 
;<br/>
<b>hence </b><a NAME="E2:46_5_3_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_5_3_1_2_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:46_5_3_1_2_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:46_5_3_1_2_2_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> 2 <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E2:46_5_3_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:46_5_3_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>

<b>hence </b><a NAME="E4:46_5_3_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:46_5_3_1_2_2_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_1_2"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> 1 <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:46_5_3_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<b>thus </b><a NAME="E2:46_5_3_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<b>hence </b><a NAME="E3:46_5_3_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/>

<b>hence </b><a NAME="E4:46_5_3_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:46_5_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:46_5_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E6:46_5_3_1_2_2_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E2:46_5_3_1_2_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:46_5_3_1_2_2_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:46_5_3_1_2_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:46"><i><font color="Green">E34</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>hence </b><a NAME="E8:46_5_3_1_2_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E11:46_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="integra1.html#E12:46_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E5:46_5_3_1_2_2_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra1.html#E6:46_5_3_1_2_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:46_5_3_1_2_2_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:46_5_3_1_2_2_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:46_5_3_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>

<b>hence </b><a NAME="E3:46_5_3_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:46_5_3_1_2_2_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_5_3_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>

<a NAME="E2:46_5_3_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:46_5_3_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E4:46_5_3_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E5:46_5_3_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:46_5_3_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:46_5_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E11:46_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E6:46_5_3_1_2_2_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_5_3_1_2_2_1_2_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>

<b>thus </b><a NAME="E2:46_5_3_1_2_2_1_2_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E3:46_5_3_1_2_2_1_2_3"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E4:46_5_3_1_2_2_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E5:46_5_3_1_2_2_1_2_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>

<b>hence </b><a NAME="E6:46_5_3_1_2_2_1_2_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E7:46_5_3_1_2_2_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:46_5_3_1_2_2_1_2"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E6:46_5_3_1_2_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E2:46_5_3_1_2_2_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra1.html#E6:46_5_3_1_2_2_1_2"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:46_5_3_1_2_2_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:46_5_3_1_2_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:46"><i><font color="Green">E34</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_2"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E9:46_5_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E10:46_5_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E11:46_5_3_1_2_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E12:46_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E5:46_5_3_1_2_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra1.html#E7:46_5_3_1_2_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="integra1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_5_3_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:46_5_3_1_2_2_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:46_5_3_1_2_2_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_3_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>hence </b><a NAME="E2:46_5_3_1_2_2_1_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E8:46_5"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>

<a NAME="E3:46_5_3_1_2_2_1_3_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:46_5_3_1_2_2_1_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E8:46_5"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:46_5_3_1_2_2_1_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_3_2"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:46_5_3_1_2_2_1_3_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:46_5_3_1_2_2_1_3_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:46_5_3_1_2_2_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:46_5_3_1_2_2_1_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E6:46_5_3_1_2_2_1_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_3_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:46_5_3_1_2_2_1_3_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E9:46_5_3_1_2_2_1_3_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:46_5_3_1_2_2_1_3_2"/><b>then </b>
not <font color="Maroon">c<sub>8</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E11:46_5_3_1_2_2_1_3_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/>
<a NAME="E12:46_5_3_1_2_2_1_3_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:46_5_3_1_2_2_1_3_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E14:46_5_3_1_2_2_1_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E8:46_5_3_1_2_2_1_3_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>

<a NAME="E15:46_5_3_1_2_2_1_3_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>thus </b><a NAME="E16:46_5_3_1_2_2_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>thus </b><a NAME="E17:46_5_3_1_2_2_1_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E15:46_5_3_1_2_2_1_3_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:46_5_3_1_2_2_1_3"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E2:46_5_3_1_2_2_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_3"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:46_5_3_1_2_2_1_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:46_5_3_1_2_2_1_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_3_1_2_2_1_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:46_5_3_1_2_2_1_3_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:46_5_3_1_2_2_1_3_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_3"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E3:46_5_3_1_2_2_1_3_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:46_5_3_1_2_2_1_3_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E5:46_5_3_1_2_2_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E6:46_5_3_1_2_2_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E7:46_5_3_1_2_2_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra1.html#E3:46_5_3_1_2_2_1_3"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:46_5_3_1_2_2_1_3_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:46_5_3_1_2_2_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:46"><i><font color="Green">E34</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E7:46_5_3_1_2_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E8:46_5_3_1_2_2_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E11:46_5"><i><font color="Green">E47</font></i></a>, <a class="txt" href="integra1.html#E12:46_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="integra1.html#E1:46_5_3_1_2_2_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra1.html#E2:46_5_3_1_2_2_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E4:46_5_3_1_2_2_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra1.html#E5:46_5_3_1_2_2_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:46_5_3_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:46_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:46"/><i><font color="Green">E40</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6"><b>proof </b></a><div class="add">

<a NAME="E1:46_6"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:46"><i><font color="Green">E39</font></i></a>, <a class="ref" href="integra1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E2:46_6"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<a NAME="E3:46_6"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T106">RVSUM_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:46_6"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:46"/><i><font color="Green">E41</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="integra1.html#E13:46"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E15:46"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_7"><b>proof </b></a><div class="add">

<a NAME="E1:46_7"/>
 <a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_7_1"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E2:46"><i><font color="Green">E33</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E2:46_7_1"/>
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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_7_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:46_7_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3" target="_self">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E3:46_7_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:46"><i><font color="Green">E33</font></i></a>, <a class="txt" href="integra1.html#E1:46_7_1_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:46_7_1"/>
 <a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:46_7_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_7"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:46"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" target="_self">lower_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3" target="_self">vol</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E14:46"><i><font color="Green">E41</font></i></a></i>;<br/>


</div>
