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

<b>let </b><font color="Maroon" title="c1">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="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="integra1.html#R3" title="INTEGRA1:pred.3">is_integrable_on</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ;<br/>

<a NAME="E2:35"/><i><font color="Green" title="E26">A2</font></i>: 
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D15" title="INTEGRA1:def.15">INTEGRA1:def 15</a></i>;<br/>
<a NAME="E3:35"/>
not  <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="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E4:35"/><b>then </b><i><font color="Green" title="E27">A3</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="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E5:35"/>
for <font color="Olive" title="b1">a</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c2">f</font></span>)</span> holds <br/><font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c2">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">D</font> being    <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="c1">A</font><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c4">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="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9" title="INTEGRA1:func.9">upper_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<i><font color="Green" title="E29">A5</font></i>: <font color="Maroon" title="c3">a</font> = 
 <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font>
<b>by </b><i><a class="txt" href="integra2.html#E3:35_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="integra1.html#D11" title="INTEGRA1:def.11">INTEGRA1:def 11</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D9" title="INTEGRA1:def.9">INTEGRA1:def 9</a></i>
;<br/>
<a NAME="E5:35_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>set </b><font color="Maroon" title="c6">r</font> = <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font>;<br/>
<b>assume </b><a NAME="E1:35_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span>
 ;<br/>

<a NAME="E2:35_1_2"/><b>then </b>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:35_1_2"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E4:35_1_2"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E5:35_1_2"/><i><font color="Green" title="E32">A8</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/>
<a NAME="E6:35_1_2"/>
 <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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2_1"/><i><font color="Green" title="E33">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2_1_1"/><i><font color="Green" title="E34">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E2:35_1_2_1_1"/><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E3:35_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:35_1_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:35_1_2_1_1"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:35_1_2_1_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:35_1_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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> 
 <b>by </b><i><a class="txt" href="integra2.html#E1:35_1_2_1_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35_1_2_1"/><i><font color="Green" title="E34">A12</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</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="35_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E2:35_1_2_1_2"/><b>then </b><i><font color="Green" title="E35">A13</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> 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/>
<a NAME="E3:35_1_2_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1_2_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</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="txt" href="integra2.html#E2:35_1_2_1_2"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="rcomp_1.html#T4" title="RCOMP_1:th.4">RCOMP_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:35_1_2_1"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c7">x</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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra2.html#E1:35_1_2_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:35_1_2_1"/>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra2.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="integra2.html#E4:35_1_2_1"><i><font color="Green" title="E35">A14</font></i></a></i>;<br/>
<a NAME="E6:35_1_2_1"/><b>then </b><i><font color="Green" title="E36">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra2.html#E4:35_1_2_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:35_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E4:35_1_2_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_2_1"/>
 <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="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra2.html#E2:35_1_2_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="integra2.html#E6:35_1_2_1"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:35_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="integra2.html#E4:35_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="integra2.html#E5:35_1_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:35_1"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra2.html#E4:35_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35"/><b>then </b>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra2.html#E2:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="integra2.html#E4:35"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="seq_4.html#T60" title="SEQ_4:th.60">SEQ_4:60</a></i>;<br/>
<b>hence </b><a NAME="E7:35"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra1.html#D18" title="INTEGRA1:def.18">INTEGRA1:def 18</a></i>;<br/>


</div>
