<?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="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:34"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R1">is_upper_integrable_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1"/>
 <a href="integra1.html#K9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1_1_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_1_1_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#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="txt" href="integra2.html#E1:34_1_1_1_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="integra2.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_1_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_1_1_2"/>
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#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="txt" href="integra2.html#E1:34_1_1_1_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="integra2.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<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="E2:34_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1_1"/>
 <a href="integra1.html#K9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font>
 <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="E2:34_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R1">is_upper_integrable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="integra1.html#D13">INTEGRA1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:34"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R2">is_lower_integrable_on</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_2_1"/>
 <a href="integra1.html#K10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rfunct_1.html#R1">is_bounded_above_on</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_2_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_2_1_1_1_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_2_1_1_1_1_1_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#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="txt" href="integra2.html#E1:34_2_1_1_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="integra2.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:34_2_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_2_1_1_1_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_2_1_1_1_1_1_2"/>
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#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="txt" href="integra2.html#E1:34_2_1_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="integra2.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E3:34_2_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<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="E2:34_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</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="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2_1_1"/>
 <a href="integra1.html#K10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rfunct_1.html#R1">is_bounded_above_on</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R2">is_lower_integrable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="integra1.html#D14">INTEGRA1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:34"/>
(  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<a NAME="E1:34_3"/><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <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#D11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E2:34_3"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</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="E3:34_3"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font>, <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="E4:34_3"/><i><font color="Green">E31</font></i>: 
not  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E5:34_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R1">is_upper_integrable_on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R2">is_lower_integrable_on</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green">E25</font></i></a>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>;<br/>
<a NAME="E6:34_3"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_4.html#V2">bounded_below</a> &amp;  <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> is <a href="seq_4.html#V1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#D13">INTEGRA1:def 13</a>, <a class="ref" href="integra1.html#D14">INTEGRA1:def 14</a></i>;<br/>
<a NAME="E7:34_3"/><i><font color="Green">E33</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <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#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E8:34_3"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</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="E9:34_3"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font>, <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="E10:34_3"/><i><font color="Green">E36</font></i>: 
not  <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> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="integra2.html#E7:34_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E11:34_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_3_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1_1"><b>suppose </b></a><a NAME="E1:34_3_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_3_1_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="integra1.html#K9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_3_1_1_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_1_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>  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</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>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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="34_3_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</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/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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> ;<br/>
<a NAME="E3:34_3_1_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</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#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E4:34_3_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</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#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T27" target="_self">Th27</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K9">upper_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>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E2:34_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rfunct_1.html#T73">RFUNCT_1:73</a></i>
;<br/>
<b>hence </b><a NAME="E6:34_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_1_1"/>
 <a href="integra1.html#K9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:34_3_1_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="integra1.html#K10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_3_1_1_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> = 
 <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 href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_1_2"/>
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="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</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>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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="34_3_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_1_2_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</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/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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> ;<br/>
<a NAME="E3:34_3_1_1_1_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E4:34_3_1_1_1_2_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</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="txt" href="integra2.html#E1:34_3_1_1_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_1_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T29" target="_self">Th29</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E8:34_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rfunct_1.html#T73">RFUNCT_1:73</a></i>
;<br/>
<b>hence </b><a NAME="E6:34_3_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_1_2"/>
 <a href="integra1.html#K10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</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="txt" href="integra2.html#E1:34_3_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E41</font></i>:  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34_3_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="integra1.html#D15">INTEGRA1:def 15</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="integra2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</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"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="integra2.html#E6:34_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D15">INTEGRA1:def 15</a></i>
;<br/> <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="integra1.html#D16">INTEGRA1:def 16</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="integra2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E9:34_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E6:34_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="integra2.html#E10:34_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D16">INTEGRA1:def 16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green">E25</font></i></a>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>
;<br/><b>hence </b><a NAME="E6:34_3_1_1_1"/>
 <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><b>thus </b><a NAME="E7:34_3_1_1_1"/>
 <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="integra1.html#D18">INTEGRA1:def 18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_3_1_1_2"><b>suppose </b></a><a NAME="E1:34_3_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_3_1_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="integra1.html#K9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_3_1_1_2_1"/><i><font color="Green">E40</font></i>: 
 <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>;<br/>
<i><font color="Green">E41</font></i>:  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> = 
 <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 href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_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>  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</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>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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="34_3_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_2_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</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/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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> ;<br/>
<a NAME="E3:34_3_1_1_2_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</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#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_2_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E4:34_3_1_1_2_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</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="txt" href="integra2.html#E1:34_3_1_1_2_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T30" target="_self">Th30</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<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>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_2_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E8:34_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="rfunct_1.html#T73">RFUNCT_1:73</a></i>
;<br/>
<b>hence </b><a NAME="E6:34_3_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_2_1"/>
 <a href="integra1.html#K9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</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="txt" href="integra2.html#E1:34_3_1_1_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:34_3_1_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="integra1.html#K10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_3_1_1_2_2"/><i><font color="Green">E41</font></i>: 
 <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <a href="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_2_2"/>
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="integra1.html#K8">divs</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</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>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <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="34_3_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_2_2_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</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/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</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> ;<br/>
<a NAME="E3:34_3_1_1_2_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_2_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E4:34_3_1_1_2_2_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</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#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_2_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_2_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K9">upper_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>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_2_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="integra2.html#E2:34_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rfunct_1.html#T73">RFUNCT_1:73</a></i>
;<br/>
<b>hence </b><a NAME="E6:34_3_1_1_2_2_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_2_2"/>
 <a href="integra1.html#K10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E41</font></i>:  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34_3_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="integra1.html#D15">INTEGRA1:def 15</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="integra2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E9:34_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E6:34_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="integra2.html#E10:34_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T14" target="_self">Th14</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D16">INTEGRA1:def 16</a></i>
;<br/> <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="integra1.html#D16">INTEGRA1:def 16</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="integra2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</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"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="integra2.html#E6:34_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra2.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D15">INTEGRA1:def 15</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green">E25</font></i></a>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>
;<br/><b>hence </b><a NAME="E6:34_3_1_1_2"/>
 <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/> <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11">upper_integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green">E25</font></i></a>, <a class="txt" href="integra2.html#E4:34_3_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D18">INTEGRA1:def 18</a></i>
;<br/><b>hence </b><a NAME="E8:34_3_1_1_2"/>
 <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:34_3"/>
(  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="integra1.html#K11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="integra1.html#K13">integral</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13">integral</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra2.html#E3:34"><i><font color="Green">E26</font></i></a>, <a class="txt" href="integra2.html#E4:34"><i><font color="Green">E27</font></i></a>, <a class="ref" href="integra1.html#D17">INTEGRA1:def 17</a>, <a class="ref" href="integra1.html#D18">INTEGRA1:def 18</a></i>;<br/>


</div>
