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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 ;<br/>

<a NAME="E2:35"/><i><font color="Green">E38</font></i>: 
 <a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D15">INTEGRA1:def 15</a></i>;<br/>
<a NAME="E3:35"/>
not  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">f</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>;<br/>
<a NAME="E4:35"/><b>then </b><i><font color="Green">E39</font></i>: 
not  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">f</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E5:35"/>
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<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">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">D</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K9">upper_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<i><font color="Green">E42</font></i>: <font color="Maroon">a</font> = 
 <a href="integra1.html#K6">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="integra1.html#D11">INTEGRA1:def 11</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D9">INTEGRA1:def 9</a></i>
;<br/>
<a NAME="E5:35_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>;<br/>
<b>assume </b><a NAME="E1:35_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E2:35_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:35_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E4:35_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E5:35_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<a NAME="E6:35_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<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">E46</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">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">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E2:35_1_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:35_1_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:35_1_2_1_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:35_1_2_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="integra2.html#E1:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35_1_2_1"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2_1_2"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E2:35_1_2_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>
<a NAME="E3:35_1_2_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1_2_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T4">RCOMP_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:35_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:35_1_2_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:35_1_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:35_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_2_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra2.html#E1:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="integra2.html#D1" target="_self">Def1</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:35_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:35_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35"/><b>then </b>
 <a href="integra1.html#K11">upper_integral</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, , <a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/>
<b>hence </b><a NAME="E7:35"/>
 <a href="integra1.html#K13">integral</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#D18">INTEGRA1:def 18</a></i>;<br/>


</div>
