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

<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E3:34"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">is_upper_integrable_on</a> <font color="Maroon" title="c2">A</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">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" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</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" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</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" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 or <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="E27">A4</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_1_1_1"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="integra2.html#E1:34_1_1_1_1_1_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="integra2.html#T19" target="_self" title="INTEGRA2:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</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" title="E27">A5</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_1_1_1_1_1_1_2"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="integra2.html#E1:34_1_1_1_1_1_1_2"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="integra2.html#T20" target="_self" title="INTEGRA2:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</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" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1_1"/>
 <a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="integra1.html#T13" title="INTEGRA1:th.13">INTEGRA1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1"/>
<font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">is_upper_integrable_on</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="integra1.html#D13" title="INTEGRA1:def.13">INTEGRA1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:34"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integra1.html#R2" title="INTEGRA1:pred.2">is_lower_integrable_on</a> <font color="Maroon" title="c2">A</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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">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" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</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" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</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" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 or <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="E28">A7</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_2_1_1_1_1_1_1"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="integra2.html#E1:34_2_1_1_1_1_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="integra2.html#T21" target="_self" title="INTEGRA2:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E3:34_2_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</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" title="E28">A8</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_2_1_1_1_1_1_2"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="integra2.html#E1:34_2_1_1_1_1_1_2"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="integra2.html#T22" target="_self" title="INTEGRA2:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E3:34_2_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</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" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>for <font color="Olive" title="b2">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2_1_1"/>
 <a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D9" title="RFUNCT_1:def.9">RFUNCT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="integra1.html#T15" title="INTEGRA1:th.15">INTEGRA1:15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2"/>
<font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integra1.html#R2" title="INTEGRA1:pred.2">is_lower_integrable_on</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="integra1.html#D14" title="INTEGRA1:def.14">INTEGRA1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:34"/>
(  <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp;  <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">f</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" title="E28">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E2:34_3"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
 <a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E3:34_3"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
 <a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131" title="FUNCT_2:th.131">FUNCT_2:131</a></i>;<br/>
<a NAME="E4:34_3"/><i><font color="Green" title="E31">A12</font></i>: 
not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34_3"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E5:34_3"/>
( <font color="Maroon" title="c3">f</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">is_upper_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">f</font> <a href="integra1.html#R2" title="INTEGRA1:pred.2">is_lower_integrable_on</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="integra1.html#D17" title="INTEGRA1:def.17">INTEGRA1:def 17</a></i>;<br/>
<a NAME="E6:34_3"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#D13" title="INTEGRA1:def.13">INTEGRA1:def 13</a>, <a class="ref" href="integra1.html#D14" title="INTEGRA1:def.14">INTEGRA1:def 14</a></i>;<br/>
<a NAME="E7:34_3"/><i><font color="Green" title="E33">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E8:34_3"/><b>then </b><i><font color="Green" title="E34">A15</font></i>: 
 <a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E9:34_3"/><b>then </b><i><font color="Green" title="E35">A16</font></i>: 
 <a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131" title="FUNCT_2:th.131">FUNCT_2:131</a></i>;<br/>
<a NAME="E10:34_3"/><i><font color="Green" title="E36">A17</font></i>: 
not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="integra2.html#E7:34_3"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E11:34_3"/><i><font color="Green" title="E37">A18</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="integra2.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">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" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 or <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="E38">A19</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_3_1_1_1"/><i><font color="Green" title="E39">A20</font></i>: 
 <a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</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" title="E40">A21</font></i>: 
 <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<i><font color="Green" title="E41">A22</font></i>:  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_1_1"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</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" title="c4">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_1_1_2"/><i><font color="Green" title="E42">A23</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c4">D</font> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> ;<br/>
<a NAME="E3:34_3_1_1_1_1_2"/><i><font color="Green" title="E43">A24</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</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" title="E42">A23</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E4:34_3_1_1_1_1_2"/><i><font color="Green" title="E44">A25</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</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" title="E42">A23</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font> = 
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_1_1_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="integra2.html#T27" target="_self" title="INTEGRA2:th.27">Th27</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</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" title="E44">A25</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E2:34_3"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="rfunct_1.html#T73" title="RFUNCT_1:th.73">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" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_1_1"/>
 <a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_1_1"><i><font color="Green" title="E40">A21</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_1_1"><i><font color="Green" title="E41">A22</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:34_3_1_1_1"/><i><font color="Green" title="E40">A26</font></i>: 
 <a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</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" title="E41">A27</font></i>: 
 <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<i><font color="Green" title="E42">A28</font></i>:  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_1_2"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</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" title="c4">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_1_2_2"/><i><font color="Green" title="E43">A29</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c4">D</font> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> ;<br/>
<a NAME="E3:34_3_1_1_1_2_2"/><i><font color="Green" title="E44">A30</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</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" title="E43">A29</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E4:34_3_1_1_1_2_2"/><i><font color="Green" title="E45">A31</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</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" title="E43">A29</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font> = 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_1_2_2"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="integra2.html#T29" target="_self" title="INTEGRA2:th.29">Th29</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</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" title="E45">A31</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E8:34_3"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="rfunct_1.html#T73" title="RFUNCT_1:th.73">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" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_1_2"/>
 <a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_1_2"><i><font color="Green" title="E41">A27</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_1_2"><i><font color="Green" title="E42">A28</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green" title="E41">A32</font></i>:  <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</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" title="E39">A20</font></i></a>, <a class="ref" href="integra1.html#D15" title="INTEGRA1:def.15">INTEGRA1:def 15</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self" title="INTEGRA2:th.17">Th17</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="integra2.html#E6:34_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="integra2.html#T15" target="_self" title="INTEGRA2:th.15">Th15</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D15" title="INTEGRA1:def.15">INTEGRA1:def 15</a></i>
;<br/> <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</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" title="E40">A26</font></i></a>, <a class="ref" href="integra1.html#D16" title="INTEGRA1:def.16">INTEGRA1:def 16</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E9:34_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self" title="INTEGRA2:th.17">Th17</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E6:34_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="integra2.html#E10:34_3"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_1"><i><font color="Green" title="E38">A19</font></i></a>, <a class="ref" href="integra2.html#T13" target="_self" title="INTEGRA2:th.13">Th13</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D16" title="INTEGRA1:def.16">INTEGRA1:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="integra1.html#D17" title="INTEGRA1:def.17">INTEGRA1:def 17</a></i>
;<br/><b>hence </b><a NAME="E6:34_3_1_1_1"/>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_1"><i><font color="Green" title="E41">A32</font></i></a></i>;<br/><b>thus </b><a NAME="E7:34_3_1_1_1"/>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_1"><i><font color="Green" title="E41">A32</font></i></a>, <a class="ref" href="integra1.html#D18" title="INTEGRA1:def.18">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" title="E38">A33</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:34_3_1_1_2"/><i><font color="Green" title="E39">A34</font></i>: 
 <a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</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" title="E40">A35</font></i>: 
 <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<i><font color="Green" title="E41">A36</font></i>:  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_2_1"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</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" title="c4">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_2_1_2"/><i><font color="Green" title="E42">A37</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c4">D</font> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> ;<br/>
<a NAME="E3:34_3_1_1_2_1_2"/><i><font color="Green" title="E43">A38</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</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" title="E42">A37</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E4:34_3_1_1_2_1_2"/><i><font color="Green" title="E44">A39</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</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" title="E42">A37</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font> = 
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_2_1_2"><i><font color="Green" title="E43">A38</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="integra2.html#T30" target="_self" title="INTEGRA2:th.30">Th30</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</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" title="E44">A39</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E8:34_3"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="rfunct_1.html#T73" title="RFUNCT_1:th.73">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" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_2_1"/>
 <a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_2_1"><i><font color="Green" title="E40">A35</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_2_1"><i><font color="Green" title="E41">A36</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:34_3_1_1_2"/><i><font color="Green" title="E40">A40</font></i>: 
 <a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</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" title="E41">A41</font></i>: 
 <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<i><font color="Green" title="E42">A42</font></i>:  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E3:34_3_1_1_2_2"/>
for <font color="Olive" title="b1">D</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">D</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" title="c4">D</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_3_1_1_2_2_2"/><i><font color="Green" title="E43">A43</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c4">D</font> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font> ;<br/>
<a NAME="E3:34_3_1_1_2_2_2"/><i><font color="Green" title="E44">A44</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</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" title="E43">A43</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E4:34_3_1_1_2_2_2"/><i><font color="Green" title="E45">A45</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</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" title="E43">A43</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font> = 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3_1_1_2_2_2"><i><font color="Green" title="E44">A44</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E11:34_3"><i><font color="Green" title="E37">A18</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="integra2.html#T28" target="_self" title="INTEGRA2:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</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" title="E45">A45</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E2:34_3"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="rfunct_1.html#T73" title="RFUNCT_1:th.73">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" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p2">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_3_1_1_2_2"/>
 <a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:34_3_1_1_2_2"><i><font color="Green" title="E41">A41</font></i></a>, <a class="txt" href="integra2.html#E2:34_3_1_1_2_2"><i><font color="Green" title="E42">A42</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green" title="E41">A46</font></i>:  <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</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" title="E39">A34</font></i></a>, <a class="ref" href="integra1.html#D15" title="INTEGRA1:def.15">INTEGRA1:def 15</a></i>
<br/>.= 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E9:34_3"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self" title="INTEGRA2:th.17">Th17</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E6:34_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="integra2.html#E10:34_3"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="integra2.html#T14" target="_self" title="INTEGRA2:th.14">Th14</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D16" title="INTEGRA1:def.16">INTEGRA1:def 16</a></i>
;<br/> <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <span class="p3">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</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" title="E40">A40</font></i></a>, <a class="ref" href="integra1.html#D16" title="INTEGRA1:def.16">INTEGRA1:def 16</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="integra2.html#K1" title="INTEGRA2:func.1">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E3:34_3"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="integra2.html#T17" target="_self" title="INTEGRA2:th.17">Th17</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E4:34_3"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="integra2.html#E6:34_3"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="integra2.html#E1:34_3_1_1_2"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="integra2.html#T16" target="_self" title="INTEGRA2:th.16">Th16</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D15" title="INTEGRA1:def.15">INTEGRA1:def 15</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="integra1.html#D17" title="INTEGRA1:def.17">INTEGRA1:def 17</a></i>
;<br/><b>hence </b><a NAME="E6:34_3_1_1_2"/>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E4:34_3_1_1_2"><i><font color="Green" title="E41">A46</font></i></a></i>;<br/> <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> = 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="txt" href="integra2.html#E2:34"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="integra2.html#E4:34_3_1_1_2"><i><font color="Green" title="E41">A46</font></i></a>, <a class="ref" href="integra1.html#D17" title="INTEGRA1:def.17">INTEGRA1:def 17</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D18" title="INTEGRA1:def.18">INTEGRA1:def 18</a></i>
;<br/><b>hence </b><a NAME="E8:34_3_1_1_2"/>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">f</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" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp;  <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
( <font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra2.html#E3:34"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="integra2.html#E4:34"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="integra1.html#D17" title="INTEGRA1:def.17">INTEGRA1:def 17</a>, <a class="ref" href="integra1.html#D18" title="INTEGRA1:def.18">INTEGRA1:def 18</a></i>;<br/>


</div>
