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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="integra1.html#V1">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>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E20</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:18"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:18"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</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#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E7:18"/><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>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green">E23</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#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:18"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E10:18"/><b>then </b><i><font color="Green">E24</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>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> or <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#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="integra3.html#E10:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="integra1.html#T8">INTEGRA1:8</a></i>;<br/><a NAME="E3:18_1_1_1"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> <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="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E4:18_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green">E25</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#K3">inf</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</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> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E2:18_1_1_2_1"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</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="txt" href="integra3.html#E6:18"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:18_1_1_2_1"/>
 <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<a NAME="E4:18_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18"><i><font color="Green">E20</font></i></a>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E5:18_1_1_2_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E6:18_1_1_2_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E10:18"><i><font color="Green">E24</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> 1 as  <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="integra1.html#T36">INTEGRA1:36</a></i>;<br/>
<a NAME="E8:18_1_1_2_1"/><i><font color="Green">E29</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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> <a href="real_1.html#K5">-</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E9:18_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E11:18_1_1_2_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T36">FINSEQ_5:36</a>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E12:18_1_1_2_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:18_1_1_2_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E14:18_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <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>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E15:18_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="integra3.html#E13:18_1_1_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E16:18_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is    <a href="integra1.html#M1">DivisionPoint</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:18_1_1_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="integra1.html#D4">INTEGRA1:def 4</a></i>;<br/>
<a NAME="E18:18_1_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E19:18_1_1_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E20:18_1_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>1</sub></font> = 
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="integra3.html#E4:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>
;<br/>
<a NAME="E2:18_1_1_2_1_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_2_1_1_2"/><i><font color="Green">E36</font></i>: 
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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E2:18_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E37</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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E3:18_1_1_2_1_1_2"/><i><font color="Green">E38</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_2"/>
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#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</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> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</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="integra3.html#E3:18_1_1_2_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1</span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E36</font></i>:  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E10:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E10:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green">E25</font></i></a></i>
;<br/>
<a NAME="E4:18_1_1_2_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <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>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18"><i><font color="Green">E23</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_4"><b>proof </b></a><div class="add">

<i><font color="Green">E38</font></i>: 2 <a href="binarith.html#K5">-'</a> 1 = 
2 <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
1

;<br/>
<a NAME="E2:18_1_1_2_1_1_4"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:18_1_1_2_1_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_4"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="jordan3.html#T26">JORDAN3:26</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1_4"/><i><font color="Green">E41</font></i>: 
( 2 <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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; 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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E6:18_1_1_2_1_1_4"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_4_2"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1

;<br/>
<b>hence </b><a NAME="E2:18_1_1_2_1_1_4_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:18_1_1_2_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="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></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"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</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="18_1_1_2_1_1_4_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_1_1_2_1_1_4_3"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:18_1_1_2_1_1_4_3"/><b>then </b><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:18_1_1_2_1_1_4_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 )
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_4_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1 )
 
;<br/>
<b>then </b><i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</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>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>
<a NAME="E6:18_1_1_2_1_1_4_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</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="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:18_1_1_2_1_1_4_3"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</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#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:18_1_1_2_1_1_4_3"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<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>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E9:18_1_1_2_1_1_4_3"/><i><font color="Green">E48</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_2_1_1_4_3_2"/><i><font color="Green">E49</font></i>: 
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:18_1_1_2_1_1_4_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</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>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:18_1_1_2_1_1_4_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:18_1_1_2_1_1_4_3_2"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</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>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</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>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_4_3_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1_4_3_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_2_1_1_4_3_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_4_3_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_4_3_2_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4_3_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:18_1_1_2_1_1_4_3_2_1_1"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</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"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="integra3.html#E3:18_1_1_2_1_1_4_3_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4_3_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3_2_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2_1_1_4_3_2_1_1"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3_2_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_4_3_2_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_4_3_2_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4_3_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:18_1_1_2_1_1_4_3_2_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4_3_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> = 
<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>8</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>

;<br/><a NAME="E5:18_1_1_2_1_1_4_3_2_1_2"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</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"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</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">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:18_1_1_2_1_1_4_3_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4_3_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><b>hence </b><a NAME="E6:18_1_1_2_1_1_4_3_2_1_2"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:18_1_1_2_1_1_4_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E11:18_1_1_2_1_1_4_3"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,2,<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">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1_1_4_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="integra3.html#E9:18_1_1_2_1_1_4_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1_1_2_1_1_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> 1 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra3.html#E6:18_1_1_2_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><font color="Maroon">c<sub>1</sub></font> = 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="integra3.html#E3:18_1_1_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra3.html#E4:18_1_1_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="rvsum_1.html#T106">RVSUM_1:106</a></i>
<br/>.= 
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E7:18_1_1_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="integra3.html#E19:18_1_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:18_1_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E22:18_1_1_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/>
<b>then </b><i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="goboard1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 2

;<br/>
<a NAME="E24:18_1_1_2_1"/>
1 <a href="nat_1.html#K1">+</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="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E25:18_1_1_2_1"/><b>then </b>
2 <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_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:18_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</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> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E10:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E27:18_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="integra3.html#E19:18_1_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="integra3.html#E20:18_1_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="integra3.html#E23:18_1_1_2_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:18"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
