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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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>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><b>that </b><br/><a NAME="E1:24"/><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>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<a NAME="E3:24"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E4:24"/>
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T67">RFUNCT_1:67</a></i>;<br/>
<a NAME="E5:24"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131">FUNCT_2:131</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#T97">RFUNCT_1:97</a></i>;<br/>
<a NAME="E7:24"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <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="E8:24"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> &amp; <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="E9:24"/><b>then </b><i><font color="Green">E44</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="E10:24"/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E11:24"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font>
 
<b>by </b><i>, , <a class="ref" href="integra1.html#T30">INTEGRA1:30</a></i>;<br/>
<a NAME="E12:24"/><i><font color="Green">E46</font></i>: 
 <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="integra1.html#T29">INTEGRA1:29</a></i>;<br/>
 <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">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="integra2.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E14:24"/>
<span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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