<?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="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:7"/><i><font color="Green">E38</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:7"/>
<span class="p1">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E3:7"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">D1</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>  st <font color="Olive">D1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">D1</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D1</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:7_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">D1</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D1</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font>
<b>by </b><i>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
;<br/>
<a NAME="E4:7_1"/>
 <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<i><font color="Green">E43</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
1
<b>by </b><i>, </i>
;<br/>
<a NAME="E2:7_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E3:7_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/>
<a NAME="E4:7_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2"><b>proof </b></a><div class="add">

<i><font color="Green">E49</font></i>:  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span> = 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>
;<br/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,1 = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E3:7_1_2_2"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:7_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><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">D1</font>,1</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> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E7:7_1_2"/>
 <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:7"/><b>then </b>
 <a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font> <a href="partfun2.html#R1">is_constant_on</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="partfun2.html#D3">PARTFUN2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:7"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="partfun2.html#T56">PARTFUN2:56</a></i>;<br/>
<a NAME="E8:7"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E9:7"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<a NAME="E1:7_2"/>
( <font color="Maroon">x1</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#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> implies <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_2_1"/>
<font color="Maroon">x1</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#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">D1</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="E3:7_2_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:7_2_1"/>
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E5:7_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:7_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E7:7_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_2"/>
(  not <font color="Maroon">x1</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#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:7"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:7"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T15">SEQ_4:15</a></i>;<br/>
<a NAME="E12:7"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:7"/>
<font color="Maroon">f</font> <a href="integra1.html#R2">is_lower_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="integra1.html#D14">INTEGRA1:def 14</a></i>;<br/>

<a NAME="E14:7"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>proof </b></a><div class="add">

<a NAME="E1:7_3"/>
0 <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#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">D1</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>
<a NAME="E2:7_3_1"/>
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>
 
;<br/>
<a NAME="E3:7_3_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E4:7_3_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K8">divs</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:7_3_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D1</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#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:7_3_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:7_3_1"/>
0 <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#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:7"/><b>then </b>
 <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"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>;<br/>
<b>hence </b><a NAME="E16:7"/>
 <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#D16">INTEGRA1:def 16</a></i>;<br/>


</div>
