<?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   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:69"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <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> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> implies  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> );<br/>
<a NAME="E1:69_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_1"/><i><font color="Green">E53</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:69_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
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>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:69_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:69_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E5:69_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E6:69_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:69_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:69_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> <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"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_1_1_1"/><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> )
 ;<br/>

<a NAME="E2:69_1_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E5:69_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:69_1_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:69_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E4:69_1_1_1_1"/>
<font color="Maroon">c<sub>6</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>4</sub></font> <a href="finseq_1.html#K17">|</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:69_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</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>4</sub></font> <a href="finseq_1.html#K17">|</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:69_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:69_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</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>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E8:69_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1</span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_1_1_1"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:69_1_1"/>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E8:69_1_1"/><b>then </b>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:69_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:69_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:69_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E12:69_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E13:69_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E14:69_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <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"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_1_2"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:69_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:69_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra1.html#E10:69_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E4:69_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E5:69_1_1_2"/>
<font color="Maroon">c<sub>6</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>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E13:69_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:69_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</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>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:69_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E8:69_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E9:69_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:69_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E13:69_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1;<br/>
<a NAME="E17:69_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 <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>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E18:69_1_1"/>
 <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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,1,1 as    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E17:69_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="integra1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E20:69_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E21:69_1_1"/><b>then </b>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:69_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 &amp;  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E23:69_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:69_1_1"/><b>then </b>
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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E25:69_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:69_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <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>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E17:69_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E27:69_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E28:69_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E25:69_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E29:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E27:69_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E30:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</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 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="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E31:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : (  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E32:69_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<a NAME="E33:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> as    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="integra1.html#E20:69_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra1.html#E22:69_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="integra1.html#E25:69_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="integra1.html#E26:69_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="integra1.html#T39" target="_self">Th39</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T43">PARTFUN1:43</a></i>;<br/>
<a NAME="E36:69_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E37:69_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E38:69_1_1"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1 <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E39:69_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1
 
<b>by </b><i><a class="txt" href="integra1.html#E37:69_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E40:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E41:69_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E42:69_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>14</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E2:69_1_1_3"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E3:69_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:69_1_1_3_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:69_1_1_3_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:69_1_1_3_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:69_1_1_3_1"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_3_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_1_3_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:69_1_1_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_3_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_1_3_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_1_3_1"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:69_1_1_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_3_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_1_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:69_1_1"/><b>then </b>
 <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:69_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra1.html#E41:69_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="integra1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E44:69_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 
;<br/>
<a NAME="E45:69_1_1"/><i><font color="Green">E75</font></i>: 
 <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_4"><b>proof </b></a><div class="add">

<i><font color="Green">E76</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/>
<a NAME="E2:69_1_1_4"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:69_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:69_1_1_4"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></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">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_4"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:69_1_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <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"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_1_4_2"/><i><font color="Green">E78</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:69_1_1_4_2"/><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E3:69_1_1_4_2"/><i><font color="Green">E80</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:69_1_1_4_2"/><i><font color="Green">E81</font></i>: 
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>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_4_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E5:69_1_1_4_2"/><i><font color="Green">E82</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_4_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_1_4_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:69_1_1_4_2"/><i><font color="Green">E83</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_4_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_4_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_4_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_1_4_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E84</font></i>: <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_4_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>10</sub></font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>10</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1_4_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
;<br/>
<i><font color="Green">E85</font></i>:  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>10</sub></font>,1 = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>10</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>10</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>10</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E5:69_1_1_4_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E5:69_1_1_4_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E6:69_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E6:69_1_1_4_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<i><font color="Green">E86</font></i>:  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1 = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
;<br/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <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"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
;<br/>
<b>then </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E3:69_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T9">FINSEQ_1:9</a></i>
;<br/>
<a NAME="E12:69_1_1_4_2"/><b>then </b><i><font color="Green">E87</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_4_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1_4_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E14:69_1_1_4_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E36:69_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_1_4_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_1_4_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="integra1.html#E9:69_1_1_4_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_1_1_4"/>
 <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E3:69_1_1_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:69_1_1"/>
 <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1
 
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E2:69_1_1_5"/>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:69_1_1_5"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></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">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:69_1_1_5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <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"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_1_5_1"/><i><font color="Green">E78</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:69_1_1_5_1"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</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>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_1_1_5_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>

<b>hence </b><a NAME="E2:69_1_1_5_1_1"/>
<font color="Maroon">c<sub>14</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>11</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="E3:69_1_1_5_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</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="E3:69_1_1_5_1"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_1_1_5_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a></i>;<br/>

<a NAME="E2:69_1_1_5_1_2_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:69_1_1_5_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>14</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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_1_5_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_5_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="E1:69_1_1_5_1_2_2_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>14</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="69_1_1_5_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_5_1_2_2_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_1_5_1_2_2_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E15:69_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E3:69_1_1_5_1_2_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E4:69_1_1_5_1_2_2_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><i><font color="Green">E84</font></i>:  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_1"><i><font color="Green">E81</font></i></a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1_2_2_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
;<br/> <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1_5_1_2_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</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 class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1_5_1_2_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
;<br/><b>hence </b><a NAME="E7:69_1_1_5_1_2_2_1_1"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E5:69_1_1_5_1_2_2_1_1"><i><font color="Green">E84</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_5_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_5_1_2_2_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_1_5_1_2_2_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_2_2_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_2_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:69_1_1_5_1_2_2_1_2_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E3:69_1_1_5_1_2_2_1_2_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E4:69_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
not <font color="Maroon">c<sub>14</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="E5:69_1_1_5_1_2_2_1_2_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>14</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>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="binop_2.html#K10">-</a> 0 )
 
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_2.html#T31">NAT_2:31</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E6:69_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:69_1_1_5_1_2_2_1_2_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>14</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> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1_2_2_1_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:69_1_1_5_1_2_2_1_2_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E22:69_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_1_5_1_2_2_1_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E9:69_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1_2_2_1_2_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:69_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:69_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:69_1_1_5_1_2_2_1_2_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:69_1_1_5_1_2_2_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E13:69_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra1.html#E14:69_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
;<br/>
<b>hence </b><a NAME="E3:69_1_1_5_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:69_1_1_5_1_2_2_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_2_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_2_2_1_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_2_2_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_2_2_1_2_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E22:69_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:69_1_1_5_1_2_2_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>14</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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:69_1_1_5_1_2_2_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:69_1_1_5_1_2_2_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:69_1_1_5_1_2_2_1_2_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">c<sub>11</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E13:69_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra1.html#E14:69_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
;<br/>
<b>hence </b><a NAME="E3:69_1_1_5_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_2_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E84</font></i>:  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1_2_2_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_1_5_1_2_2_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
;<br/> <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_2_2_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
;<br/><b>hence </b><a NAME="E6:69_1_1_5_1_2_2_1_2"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1_5_1_2_2_1_2"><i><font color="Green">E84</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_1_1_5_1_2"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:69_1_1_5_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E5:69_1_1_5_1"/><i><font color="Green">E81</font></i>: 
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_3"/><i><font color="Green">E82</font></i>: 
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E2:69_1_1_5_1_3"/><i><font color="Green">E83</font></i>: 
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:69_1_1_5_1_3"/>
(  <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_3_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_1_1_5_1_3_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a></i>;<br/>

<a NAME="E2:69_1_1_5_1_3_1_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:69_1_1_5_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_1_5_1_3_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_3_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_5_1_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1_5_1_3_1_2_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>14</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="69_1_1_5_1_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_5_1_3_1_2_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_1_5_1_3_1_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E15:69_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E3:69_1_1_5_1_3_1_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E4:69_1_1_5_1_3_1_2_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:69_1_1_5_1_3_1_2_1_1"/><b>then </b>
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <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#E1:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:69_1_1_5_1_3_1_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <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>5</sub></font>,<font color="Maroon">c<sub>14</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>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E7:69_1_1_5_1_3_1_2_1_1"/>
 <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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_3_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_3_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</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>4</sub></font>,1</span>)</span> &amp; <font color="Olive">b<sub>2</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>4</sub></font>,1</span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_5_1_3_1_2_1_1_1"/>
 <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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:69_1_1_5_1_3_1_2_1_1"/>
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra1.html#E6:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E9:69_1_1_5_1_3_1_2_1_1"/><i><font color="Green">E88</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E10:69_1_1_5_1_3_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E20:69_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra1.html#E22:69_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E11:69_1_1_5_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/><a NAME="E12:69_1_1_5_1_3_1_2_1_1"/><b>then </b><i><font color="Green">E89</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E9:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E13:69_1_1_5_1_3_1_2_1_1"/>
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_3_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_3_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_5_1_3_1_2_1_1_2"/>
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E15:69_1_1_5_1_3_1_2_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>15</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> &amp;  <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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E6:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E16:69_1_1_5_1_3_1_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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="integra1.html#E15:69_1_1_5_1_3_1_2_1_1"><i><font color="Green">E90</font></i></a></i>;<br/><b>hence </b><a NAME="E17:69_1_1_5_1_3_1_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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_5_1_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:69_1_1_5_1_3_1_2_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E85</font></i></a></i>;<br/><a NAME="E5:69_1_1_5_1_3_1_2_1_2"/><b>then </b>
not <font color="Maroon">c<sub>14</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="E6:69_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>14</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>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <a href="binop_2.html#K10">-</a> 0 )
 <b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_2.html#T31">NAT_2:31</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:69_1_1_5_1_3_1_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="integra1.html#E22:69_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E8:69_1_1_5_1_3_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E9:69_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a> )
 <b>by </b><i><a class="ref" href="integra1.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E11:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E90</font></i>: 
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E12:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E91</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E13:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E92</font></i>: 
 <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>4</sub></font>,1</span>)</span> <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#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E14:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E93</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>4</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E15:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E94</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="integra1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E16:69_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:69_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E10:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="integra1.html#E11:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="integra1.html#E13:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="integra1.html#E15:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E18:69_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E20:69_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra1.html#E6:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="integra1.html#E9:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E19:69_1_1_5_1_3_1_2_1_2"/><b>then </b>
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E11:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="integra1.html#E14:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E21:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>16</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> &amp;  <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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E17:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E22:69_1_1_5_1_3_1_2_1_2"/>
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="integra1.html#E21:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E96</font></i></a></i>;<br/><b>hence </b><a NAME="E23:69_1_1_5_1_3_1_2_1_2"/>
 <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E24:69_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E25:69_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green">E97</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E10:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="integra1.html#E13:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="integra1.html#E15:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E26:69_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E20:69_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1_5_1_3_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E27:69_1_1_5_1_3_1_2_1_2"/><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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="integra1.html#E14:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E29:69_1_1_5_1_3_1_2_1_2"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>17</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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> &amp;  <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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E25:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E97</font></i></a></i>;<br/><a NAME="E30:69_1_1_5_1_3_1_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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">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>4</sub></font>,1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</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>4</sub></font>,1</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="integra1.html#E29:69_1_1_5_1_3_1_2_1_2"><i><font color="Green">E98</font></i></a></i>;<br/><b>hence </b><a NAME="E31:69_1_1_5_1_3_1_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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_1_1_5_1_3_1"/>
(  <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp;  <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>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_1_1_5_1_3"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,1
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69_1_1_5_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>14</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>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_5_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_5_1_4"/><i><font color="Green">E83</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span> <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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_5_1_4"/>
<font color="Maroon">c<sub>14</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a></i>;<br/>

 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_1_5_1_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E4:69_1_1_5_1_4"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E12:69_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_1_5_1"><i><font color="Green">E79</font></i></a></i>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E6:69_1_1_5_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E6:69_1_1_5_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E8:69_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:69_1_1_5_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_1_5_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_1_1_5"/>
 <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:69_1_1_5"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E47:69_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E44:69_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="integra1.html#E45:69_1_1"><i><font color="Green">E75</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:69_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> implies  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:69_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</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>4</sub></font>
 ;<br/>

<a NAME="E3:69_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>6</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>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:69_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_2_1_1"/>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 or 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_1"/>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_2_1_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_2"/><i><font color="Green">E58</font></i>: 
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1;<br/><a NAME="E2:69_1_2_1_1_2"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/><a NAME="E3:69_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <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/><a NAME="E4:69_1_2_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E5:69_1_2_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:69_1_2_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</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#E4:69_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:69_1_2_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>6</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>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:69_1_2_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:69_1_2_1_1_2"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E10:69_1_2_1_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E11:69_1_2_1_1_2"/><i><font color="Green">E66</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/><a NAME="E12:69_1_2_1_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/><a NAME="E13:69_1_2_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:69_1_2_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp;  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:69_1_2_1_1_2"/><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp;  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E10:69_1_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="integra1.html#E13:69_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:69_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_2_1_1_2_1"/>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:69_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E5:69_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E6:69_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:69_1_2_1_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_2_1_1_2_2"/>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:69_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E5:69_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:69_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E20:69_1_2_1_1_2"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E6:69_1_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E21:69_1_2_1_1_2"/>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/><a NAME="E22:69_1_2_1_1_2"/><b>then </b>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</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>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:69_1_2_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E10:69_1_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E24:69_1_2_1_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</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="E26:69_1_2_1_1_2"/><i><font color="Green">E77</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/><a NAME="E27:69_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E15:69_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</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="E29:69_1_2_1_1_2"/><i><font color="Green">E78</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="integra1.html#E19:69_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/><a NAME="E30:69_1_2_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_3"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E20:69_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="integra1.html#E26:69_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:69_1_2_1_1_2_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Maroon">c<sub>15</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"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2_1_1_2_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> )
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>19</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>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_3_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>19</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">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_3_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:69_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<i><font color="Green">E85</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E5:69_1_2_1_1_2_3_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<i><font color="Green">E86</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E20:69_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E8:69_1_2_1_1_2_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E9:69_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E11:69_1_2_1_1_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>19</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">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2_1_1_2_3_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_3_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="integra1.html#E9:69_1_2_1_1_2_3_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E12:69_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E12:69_1_2_1_1_2_3_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E14:69_1_2_1_1_2_3_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E26:69_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_3_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="integra1.html#E6:69_1_2_1_1_2_3_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_1_2_1_1_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E31:69_1_2_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_4"/><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E23:69_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="integra1.html#E29:69_1_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:69_1_2_1_1_2_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> holds <br/><font color="Maroon">c<sub>17</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"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2_1_1_2_4_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>19</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>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_4_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>19</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">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_4"><i><font color="Green">E81</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_4_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:69_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<i><font color="Green">E86</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E5:69_1_2_1_1_2_4_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<i><font color="Green">E87</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E23:69_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E8:69_1_2_1_1_2_4_1"/><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E17:69_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E15:69_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E10:69_1_2_1_1_2_4_1"/><b>then </b>
<font color="Maroon">c<sub>19</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">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2_1_1_2_4_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_4_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2_4_1"><i><font color="Green">E88</font></i></a></i>;<br/>
<a NAME="E11:69_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E11:69_1_2_1_1_2_4_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E13:69_1_2_1_1_2_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E29:69_1_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_4_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="integra1.html#E6:69_1_2_1_1_2_4_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:69_1_2_1_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>18</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<a NAME="E2:69_1_2_1_1_2_5"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>6</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>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>6</sub></font>
 
;<br/>
<a NAME="E2:69_1_2_1_1_2_5_1"/><b>then </b>
 <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:69_1_2_1_1_2_5_1"/>
( <font color="Maroon">c<sub>6</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>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>23</sub></font> <b>by </b><i><a class="ref" href="integra1.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_5"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E16:69_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:69_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:69_1_2_1_1_2_5"/><i><font color="Green">E85</font></i>: 
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:69_1_2_1_1_2_5"/><i><font color="Green">E86</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:69_1_2_1_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:69_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <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>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:69_1_2_1_1_2_5"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E11:69_1_2_1_1_2_5"/><i><font color="Green">E89</font></i>: 
(  <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> &amp;  <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E12:69_1_2_1_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>6</sub></font>
 
;<br/>
<a NAME="E13:69_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E90</font></i>: 
 <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E14:69_1_2_1_1_2_5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:69_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra1.html#E6:69_1_2_1_1_2_5"><i><font color="Green">E85</font></i></a>, <a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5"><i><font color="Green">E87</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E15:69_1_2_1_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra1.html#E13:69_1_2_1_1_2_5"><i><font color="Green">E90</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> as    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>23</sub></font> <b>by </b><i><a class="txt" href="integra1.html#E12:69_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5"><i><font color="Green">E87</font></i></a>, <a class="txt" href="integra1.html#E10:69_1_2_1_1_2_5"><i><font color="Green">E88</font></i></a>, <a class="txt" href="integra1.html#E11:69_1_2_1_1_2_5"><i><font color="Green">E89</font></i></a>, <a class="ref" href="integra1.html#T39" target="_self">Th39</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T43">PARTFUN1:43</a></i>;<br/>
<a NAME="E18:69_1_2_1_1_2_5"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_5_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E2:69_1_2_1_1_2_5_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_5_2"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5"><i><font color="Green">E82</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_5_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E5:69_1_2_1_1_2_5_2"/>
<font color="Maroon">c<sub>26</sub></font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E6:69_1_2_1_1_2_5_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>27</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E2:69_1_2_1_1_2_5_2_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_5_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:69_1_2_1_1_2_5_2_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_5_2_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_5_2_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_5_2_1_1"/><i><font color="Green">E96</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>28</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_2_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_5_2_1_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:69_1_2_1_1_2_5_2_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_2_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_5_2_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_2_1_1"><i><font color="Green">E96</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:69_1_2_1_1_2_5_2_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_2_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_1_2_1_1_2_5_2_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:69_1_2_1_1_2_5_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:69_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>26</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E20:69_1_2_1_1_2_5"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_3"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_5_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:69_1_2_1_1_2_5_3"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_5_3"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E26:69_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_5_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>16</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="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2_1_1_2_5_3_1"/><i><font color="Green">E96</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E26:69_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_5_3_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>27</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>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_3_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:69_1_2_1_1_2_5_3_1"/>
<font color="Maroon">c<sub>6</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E24:69_1_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E6:69_1_2_1_1_2_5_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:69_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5_3_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>then </b><i><font color="Green">E101</font></i>: <font color="Maroon">c<sub>16</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E26:69_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_3_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_3_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E3:69_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
;<br/>
<a NAME="E10:69_1_2_1_1_2_5_3_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_3_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_5_3_1_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_2_1_1_2_5_3_1_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_3"><i><font color="Green">E94</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="E11:69_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green">E102</font></i>: 
(  <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>24</sub></font>,1</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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp;  <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>24</sub></font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="goboard1.html#K1">.</a> 1 )
 
<b>by </b><i><a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E12:69_1_2_1_1_2_5_3_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E13:69_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>24</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:69_1_2_1_1_2_5_3_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E14:69_1_2_1_1_2_5_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>6</sub></font>
 
;<br/>
<a NAME="E15:69_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green">E104</font></i>: 
 <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E16:69_1_2_1_1_2_5_3_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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E17:69_1_2_1_1_2_5_3_1"/><b>then </b>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="integra1.html#E15:69_1_2_1_1_2_5_3_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E18:69_1_2_1_1_2_5_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E26:69_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_3"><i><font color="Green">E94</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_3_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_5_3_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="integra1.html#E13:69_1_2_1_1_2_5_3_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="integra1.html#E15:69_1_2_1_1_2_5_3_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E19:69_1_2_1_1_2_5_3_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5_3_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_1_2_1_1_2_5_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E3:69_1_2_1_1_2_5_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:69_1_2_1_1_2_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_4"><b>proof </b></a><div class="add">

<i><font color="Green">E95</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5"><i><font color="Green">E83</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>

;<br/>
<i><font color="Green">E96</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/>
<a NAME="E3:69_1_2_1_1_2_5_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Maroon">c<sub>18</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="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:69_1_2_1_1_2_5_4_3"/><i><font color="Green">E97</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_5_4_3"/><b>then </b>
<font color="Maroon">c<sub>27</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>18</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E99</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E15:69_1_2_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E5:69_1_2_1_1_2_5_4_3"/><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</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">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E29:69_1_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_5_4_3"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_1.html#T41">FINSEQ_1:41</a></i>;<br/>
<a NAME="E6:69_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:69_1_2_1_1_2_5_4_3"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E8:69_1_2_1_1_2_5_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</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>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5_4_3"><i><font color="Green">E100</font></i></a></i>;<br/>
<a NAME="E9:69_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:69_1_2_1_1_2_5_4_3"/><i><font color="Green">E104</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5_4_3"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:69_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E12:69_1_2_1_1_2_5_4_3"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="integra1.html#E19:69_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5"><i><font color="Green">E83</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4"><i><font color="Green">E95</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E13:69_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>27</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"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E19:69_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="integra1.html#E11:69_1_2_1_1_2_5_4_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:69_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E109</font></i>: <font color="Maroon">c<sub>18</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E29:69_1_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_5_4_3"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E6:69_1_2_1_1_2_5_4_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" target="_self">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5_4_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5_4_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>
;<br/>
<a NAME="E16:69_1_2_1_1_2_5_4_3"/><i><font color="Green">E110</font></i>: 
(  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> &amp;  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_2_1_1_2_5_4_3_3_1_1"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>27</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="69_1_2_1_1_2_5_4_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>6</sub></font>
 ;<br/><a NAME="E3:69_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green">E113</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5_4_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5_4_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:69_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E23:69_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E115</font></i>:  <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>25</sub></font>,1</span>)</span> = 
 <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>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2_5_4_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E2:69_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
;<br/><i><font color="Green">E116</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>25</sub></font>,1</span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2_5_4_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="integra1.html#E3:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E7:69_1_2_1_1_2_5_4_3_3_1_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/><i><font color="Green">E118</font></i>:  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5"><i><font color="Green">E87</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5"><i><font color="Green">E87</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:69"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self">Def21</a></i>
;<br/><b>hence </b><a NAME="E9:69_1_2_1_1_2_5_4_3_3_1_1_1"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="integra1.html#E6:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/><b>thus </b><a NAME="E10:69_1_2_1_1_2_5_4_3_3_1_1_1"/>
(  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> &amp;  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2_5_4_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:69_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E16:69_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:69_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3"><i><font color="Green">E97</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E5:69_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E10:69_1_2_1_1_2_5_4_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:69_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green">E115</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K10">-</a> 1 &amp; <font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3_3_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3"><i><font color="Green">E97</font></i></a></i>;<br/>
<a NAME="E2:69_1_2_1_1_2_5_4_3_3_1_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>27</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E3:69_1_2_1_1_2_5_4_3_3_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <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="nat_2.html#T31">NAT_2:31</a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>

<a NAME="E5:69_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:69_1_2_1_1_2_5_4_3_3_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E7:69_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E19:69_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="integra1.html#E11:69_1_2_1_1_2_5_4_3"><i><font color="Green">E105</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_2_1_1_2_5_4_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:69_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3_3_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_5_4_3_3_1_1_2_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<a NAME="E2:69_1_2_1_1_2_5_4_3_3_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font>

;<br/>
<b>hence </b><a NAME="E4:69_1_2_1_1_2_5_4_3_3_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E8:69_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_4_3_3_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:69_1_2_1_1_2_5_4_3_3_1_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:69_1_2_1_1_2_5_4_3_3_1_1_2_3"/><b>then </b>
 <a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>27</sub></font>
 
;<br/>
<a NAME="E3:69_1_2_1_1_2_5_4_3_3_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E23:69_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E4:69_1_2_1_1_2_5_4_3_3_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E5:69_1_2_1_1_2_5_4_3_3_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3"><i><font color="Green">E97</font></i></a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E118</font></i>:  <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>25</sub></font>,<font color="Maroon">c<sub>27</sub></font></span>)</span> = 
<font color="Maroon">c<sub>25</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2_5_4_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="integra1.html#E4:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="integra1.html#E6:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><i><font color="Green">E119</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>25</sub></font>,<font color="Maroon">c<sub>27</sub></font></span>)</span> = 
<font color="Maroon">c<sub>25</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2_5_4_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="integra1.html#E19:69_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="integra1.html#E5:69_1_2_1_1_2_5"><i><font color="Green">E84</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5"><i><font color="Green">E86</font></i></a>, <a class="txt" href="integra1.html#E1:69_1_2_1_1_2_5_4_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="integra1.html#E11:69_1_2_1_1_2_5_4_3"><i><font color="Green">E105</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_2_1_1_2_5_4_3"><i><font color="Green">E106</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E11:69_1_2_1_1_2_5_4_3_3_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/><i><font color="Green">E121</font></i>:  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5_4_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5_4_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="integra1.html#E8:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self">Def5</a></i>
;<br/><b>hence </b><a NAME="E13:69_1_2_1_1_2_5_4_3_3_1_1_2"/>
 <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E9:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="integra1.html#E10:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/><b>thus </b><a NAME="E14:69_1_2_1_1_2_5_4_3_3_1_1_2"/>
(  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> &amp;  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E14:69_1_2_1_1_2_5_4_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="integra1.html#E11:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="integra1.html#E12:69_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:69_1_2_1_1_2_5_4_3_3"/>
(  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> &amp;  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:69_1_2_1_1_2_5_4_3"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" target="_self">divset</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<b>hence </b><a NAME="E18:69_1_2_1_1_2_5_4_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E13:69_1_2_1_1_2_5_4_3"><i><font color="Green">E107</font></i></a>, <a class="txt" href="integra1.html#E15:69_1_2_1_1_2_5_4_3"><i><font color="Green">E109</font></i></a>, <a class="txt" href="integra1.html#E16:69_1_2_1_1_2_5_4_3"><i><font color="Green">E110</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_1_2_1_1_2_5_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E29:69_1_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="integra1.html#E2:69_1_2_1_1_2_5_4"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:69_1_2_1_1_2_5"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>24</sub></font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2_1_1_2_5_5"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_2_1_1_2_5_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E4:69_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:69_1_2_1_1_2_5_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</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="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:69_1_2_1_1_2_5_5"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>24</sub></font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="ref" href="integra1.html#T31" target="_self">Th31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:69_1_2_1_1_2_5"/><b>then </b>
 <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E18:69_1_2_1_1_2_5"><i><font color="Green">E91</font></i></a>, <a class="txt" href="integra1.html#E19:69_1_2_1_1_2_5"><i><font color="Green">E92</font></i></a>, <a class="ref" href="integra1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E24:69_1_2_1_1_2_5"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</sub></font>
 
;<br/>
<b>hence </b><a NAME="E25:69_1_2_1_1_2_5"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E20:69_1_2_1_1_2_5"><i><font color="Green">E93</font></i></a>, <a class="txt" href="integra1.html#E21:69_1_2_1_1_2_5"><i><font color="Green">E94</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:69_1_2_1_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E26:69_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/><a NAME="E34:69_1_2_1_1_2"/><i><font color="Green">E83</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E29:69_1_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/><a NAME="E35:69_1_2_1_1_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>16</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"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</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> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E33:69_1_2_1_1_2"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E36:69_1_2_1_1_2"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E32:69_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E37:69_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</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> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E38:69_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</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> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:69_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra1.html#E7:69_1_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra1.html#E30:69_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="integra1.html#E31:69_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E39:69_1_2_1_1_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E34:69_1_2_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:69_1_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:69_1"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="txt" href="integra1.html#E1:69_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra1.html#E2:69_1"><i><font color="Green">E53</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <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> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" target="_self">indx</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>


</div>
