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

<b>let </b><font color="Maroon">c<sub>1</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>assume </b><a NAME="E1:16"/><i><font color="Green">E16</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:16"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:16_1"/><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green">E19</font></i>: 
 <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> 1
 ;<br/>

<a NAME="E2:16_1_1"/>
 <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="E3:16_1_1"/><b>then </b>
not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="txt" href="integra4.html#E1:16_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E4:16_1_1"/><b>then </b><i><font color="Green">E20</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> 2
 
<b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/>
<a NAME="E5:16_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>4</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:16_1_1"/><b>then </b><i><font color="Green">E21</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> &amp; 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="txt" href="integra4.html#E4:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:16_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 2
 
<b>by </b><i><a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E8:16_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>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E6:16_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="integra1.html#T8">INTEGRA1:8</a></i>;<br/>
<a NAME="E9:16_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
(  <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="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<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="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 2 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E10:16_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E1:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E11:16_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E12:16_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra4.html#E7:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="integra4.html#E9:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E19</font></i>: <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>
<br/>.= 
 <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D19">INTEGRA1:def 19</a></i>
;<br/>
<a NAME="E4:16_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:16_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>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_3_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</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#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:16_1_3_1"/><i><font color="Green">E21</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> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></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> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="integra4.html#E1:16_1_3_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E5:16_1_3_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"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:16_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:16_1_3_1"/><b>then </b><i><font color="Green">E22</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="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E7:16_1_3_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <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>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:16_1_3_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<a NAME="E8:16_1_3_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</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="E9:16_1_3_1"/>
<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="txt" href="integra4.html#E6:16_1_3_1"><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="E10:16_1_3_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</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#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E11:16_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E1:16"><i><font color="Green">E16</font></i></a>, <a class="txt" href="integra4.html#E7:16_1_3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="integra4.html#E8:16_1_3_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="integra2.html#T40">INTEGRA2:40</a></i>;<br/>
<b>hence </b><a NAME="E12:16_1_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:16_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"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> holds <br/><font color="Olive">b<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#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_3_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:16_1_3_2"/><i><font color="Green">E22</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="txt" href="integra4.html#E2:16_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E3:16_1_3_2"/><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> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E4:16_1_3_2"/><b>then </b>
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#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>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="E5:16_1_3_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <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#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:16_1_3_2"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i><a class="txt" href="integra4.html#E2:16_1_3_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<a NAME="E7:16_1_3_2"/>
(  <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>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> &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>,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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1_3_2_1"/>
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="integra4.html#E2:16_1_3_2"><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="E2:16_1_3_2_1"/><b>then </b>
(  <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>,1</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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></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>,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="integra4.html#E2:16_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_3_2_1"/>
(  <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>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">c<sub>1</sub></font> &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>,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="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <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> <font color="Maroon">c<sub>1</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>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="integra4.html#E1:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E9:16_1_3_2"/>
<font color="Maroon">c<sub>5</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#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E1:16_1_3_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="integra4.html#E5:16_1_3_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="integra4.html#E6:16_1_3_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:16_1_3"/><b>then </b>
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="integra4.html#E2:16_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:16_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:16_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:16"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D6">SEQM_3:def 6</a></i>;<br/>
<a NAME="E5:16"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39">SEQ_4:39</a></i>;<br/>
<a NAME="E6:16"/>
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E3:16"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E4:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="seq_4.html#T40">SEQ_4:40</a></i>;<br/>
<b>hence </b><a NAME="E8:16"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="integra2.html#K2">delta</a> <font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="integra4.html#E5:16"><i><font color="Green">E19</font></i></a></i>;<br/>


</div>
