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

<b>let </b><font color="Maroon">a</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> , <font color="Maroon">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> , <font color="Maroon">h</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><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E38</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="E2:34"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:34"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">g</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:34"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">h</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:34"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:34"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</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> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E8:34"/>
for <font color="Olive">T</font> being  <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>  st  <a href="integra2.html#K2">delta</a> <font color="Olive">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Olive">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Olive">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T</font> be   <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:34_1"/><i><font color="Green">E48</font></i>: 
 <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E2:34_1"/><i><font color="Green">E49</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:34_1"/><i><font color="Green">E50</font></i>: 
(  <a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i>, <a class="txt" href="integra4.html#E3:34"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra4.html#E4:34"><i><font color="Green">E41</font></i></a>, , </i>;<br/>
<a NAME="E4:34_1"/><i><font color="Green">E51</font></i>: 
(  <a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i>, <a class="txt" href="integra4.html#E3:34"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra4.html#E4:34"><i><font color="Green">E41</font></i></a>, , </i>;<br/>
<a NAME="E5:34_1"/><i><font color="Green">E52</font></i>: 
(  <a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i>, <a class="txt" href="integra4.html#E3:34"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra4.html#E4:34"><i><font color="Green">E41</font></i></a>, , </i>;<br/>
<a NAME="E6:34_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="txt" href="integra4.html#E3:34"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra4.html#E4:34"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E7:34_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="integra4.html#T1" target="_self">Th1</a>, <a class="txt" href="integra4.html#E3:34"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra4.html#E4:34"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E8:34_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="integra4.html#E6:34"><i><font color="Green">E43</font></i></a>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<a NAME="E9:34_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:34"><i><font color="Green">E42</font></i></a>, <a class="ref" href="seq_2.html#T25">SEQ_2:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">osc</font> = <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>reconsider </b><font color="Maroon">osc1</font> = <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>reconsider </b><font color="Maroon">osc2</font> = <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E13:34_1"/><i><font color="Green">E57</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E6:34"><i><font color="Green">E43</font></i></a>, <a class="ref" href="integra4.html#T2" target="_self">Th2</a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>
<a NAME="E14:34_1"/><i><font color="Green">E58</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E5:34"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>
<a NAME="E15:34_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:34_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:34_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E1:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E3:34_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:34_1_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i>, <a class="txt" href="integra4.html#E2:34_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:34_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 <b>by </b><i><a class="txt" href="integra4.html#E1:34_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="integra4.html#E3:34_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="integra4.html#E5:34_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E8:34_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">n</font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font>;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/>
<a NAME="E2:34_1_1_1"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1_1"/>
( <font color="Maroon">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:34_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E11:34_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:34_1_1_2"/><b>then </b>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E8:34_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:34_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 )
 
<b>by </b><i><a class="txt" href="integra4.html#E6:34_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra4.html#E7:34_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> ;<br/>
<a NAME="E5:34_1_1_2"/>
<font color="Maroon">osc</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<b>then </b><i><font color="Green">E65</font></i>: <font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D9">INTEGRA1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</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> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
;<br/>
<a NAME="E7:34_1_1_2"/>
<font color="Maroon">osc1</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<b>then </b><i><font color="Green">E66</font></i>: <font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">g</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">g</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D9">INTEGRA1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
;<br/>
<a NAME="E9:34_1_1_2"/>
 <a href="finseq_1.html#K3">len</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="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">UV</font> =  <a href="integra1.html#K14">upper_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E11:34_1_1_2"/>
 <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">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">LV</font> =  <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E13:34_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">UV1</font> =  <a href="integra1.html#K14">upper_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E15:34_1_1_2"/>
 <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">g</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">LV1</font> =  <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E17:34_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E13:34_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="rvsum_1.html#T120">RVSUM_1:120</a></i>;<br/>
<a NAME="E18:34_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E14:34_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rvsum_1.html#T120">RVSUM_1:120</a></i>;<br/>
<a NAME="E19:34_1_1_2"/>
<font color="Maroon">osc2</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K9">+</a> <span class="p1">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T15">SEQ_1:15</a></i>;<br/>
<b>then </b><i><font color="Green">E90</font></i>: <font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17">-</a> <span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">h</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D4">INTEGRA2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K6">upper_sum</a> <font color="Maroon">h</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">h</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">h</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">h</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D9">INTEGRA1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">h</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">h</font>,<font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>
;<br/>
<a NAME="E21:34_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <font color="Maroon">h</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">UV2</font> =  <a href="integra1.html#K14">upper_volume</a> <font color="Maroon">h</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E23:34_1_1_2"/>
 <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">h</font>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">LV2</font> =  <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">h</font>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="catalg_1.html#T5">CATALG_1:5</a></i>;<br/>
<a NAME="E25:34_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">UV2</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E5:34_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rvsum_1.html#T120">RVSUM_1:120</a></i>;<br/>
<a NAME="E26:34_1_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</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> holds <br/><span class="p1">(<span class="default"><font color="Maroon">UV2</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_2_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">j</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>
 ;<br/>

<b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><font color="Maroon">UV2</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E2:34_1_1_2_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">UV2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</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">h</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">j</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E3:34_1_1_2_4"/>
<font color="Maroon">LV2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <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">h</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">j</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><font color="Maroon">UV2</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<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="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E3:34_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></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">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E5:34_1_1_2_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">LV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <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">D</font>,<font color="Maroon">j</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E6:34_1_1_2_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">UV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</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">g</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">j</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E7:34_1_1_2_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">LV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <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">g</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">j</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T67">RVSUM_1:67</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T27">RVSUM_1:27</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">LV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">LV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">LV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T48">RVSUM_1:48</a></i>
;<br/>
<b>then </b><i><font color="Green">E98</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra4.html#E8:34_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra4.html#E17:34_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="integra4.html#T3" target="_self">Th3</a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p0">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></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">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E10:34_1_1_2_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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">h</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">h</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_4_4"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_2_4_4"/>
<font color="Maroon">j</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="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:34_1_1_2_4_4"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<i><font color="Green">E102</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">j</font></span>)</span></span>)</span> = 
<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">j</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">A</font> <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">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="integra4.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T28">PARTFUN1:28</a></i>;<br/>
<i><font color="Green">E103</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</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">j</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">A</font> <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">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="integra4.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T28">PARTFUN1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="integra4.html#E6"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E104</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</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">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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">j</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">A</font> <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">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="integra4.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T28">PARTFUN1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h1</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="integra4.html#E7"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:34_1_1_2_4_4"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_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="E12:34_1_1_2_4_4"/><b>then </b>
( <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="rfunct_1.html#R1">is_bounded_above_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="integra4.html#T5" target="_self">Th5</a>, <a class="ref" href="integra2.html#T5">INTEGRA2:5</a>, <a class="ref" href="integra2.html#T6">INTEGRA2:6</a></i>;<br/>
<a NAME="E13:34_1_1_2_4_4"/><b>then </b>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="rfunct_1.html#R3">is_bounded_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E15:34_1_1_2_4_4"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</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"><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">j</font></span>)</span></span>)</span></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><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">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T90">RFUNCT_1:90</a></i>;<br/>
<a NAME="E16:34_1_1_2_4_4"/>
( <font color="Maroon">g</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="rfunct_1.html#R2">is_bounded_below_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="E17:34_1_1_2_4_4"/><b>then </b>
( <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="rfunct_1.html#R1">is_bounded_above_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="rfunct_1.html#R2">is_bounded_below_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="integra4.html#E6"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="integra2.html#T5">INTEGRA2:5</a>, <a class="ref" href="integra2.html#T6">INTEGRA2:6</a></i>;<br/>
<a NAME="E18:34_1_1_2_4_4"/><b>then </b>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="rfunct_1.html#R3">is_bounded_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">r2</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E20:34_1_1_2_4_4"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</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"><font color="Maroon">g</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">j</font></span>)</span></span>)</span></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</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">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T90">RFUNCT_1:90</a></i>;<br/>
<a NAME="E21:34_1_1_2_4_4"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="integra4.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E22:34_1_1_2_4_4"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">f1</font> <a href="rfunct_1.html#R3">is_bounded_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T90">RFUNCT_1:90</a></i>;<br/>
<a NAME="E23:34_1_1_2_4_4"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E24:34_1_1_2_4_4"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">g1</font> <a href="rfunct_1.html#R3">is_bounded_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T90">RFUNCT_1:90</a></i>;<br/>
<a NAME="E25:34_1_1_2_4_4"/>
( <font color="Maroon">f1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">g1</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">h1</font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i><a class="ref" href="integra4.html#T5" target="_self">Th5</a>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E26:34_1_1_2_4_4"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">f1</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>, <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">g1</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>, <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">h1</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</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="E27:34_1_1_2_4_4"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_4_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:34_1_1_2_4_4_4"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> )
 ;<br/>

<a NAME="E2:34_1_1_2_4_4_4"/><b>then </b>
( <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="integra4.html#T5" target="_self">Th5</a>, <a class="txt" href="integra4.html#E6"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="integra4.html#E7"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1_2_4_4_4"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E2:34"><i><font color="Green">E39</font></i></a>, <a class="ref" href="integra4.html#T5" target="_self">Th5</a>, <a class="ref" href="integra4.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E28:34_1_1_2_4_4"/>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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">h</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">h</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra4.html#E1:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="integra4.html#T8" target="_self">Th8</a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:34_1_1_2_4"/>
 <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">j</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/>
<b>hence </b><a NAME="E12:34_1_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">UV2</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="integra4.html#E6:34_1_1_2"><i><font color="Green">E65</font></i></a>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:34_1_1_2"/><b>then </b>
<font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E7:34_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>
<a NAME="E28:34_1_1_2"/><b>then </b>
<font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>;<br/>
<a NAME="E29:34_1_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E15:34_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra4.html#E3:34_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">UV2</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E31:34_1_1_2"/>
<font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_2_5"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_2_5_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E2:34_1_1_2_5_1"/>
<font color="Maroon">j</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">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E3:34_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:34_1_1_2_5_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">j</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="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:34_1_1_2_5_1"/><b>then </b>
( <font color="Maroon">UV2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</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">h</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">j</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">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">LV2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <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">h</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">j</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">j</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E138</font></i>: <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<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="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E3:34_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></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">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E6:34_1_1_2_5_1"/><i><font color="Green">E139</font></i>: 
<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">h</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">h</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">j</font></span>)</span></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="34_1_1_2_5_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_2_5_1_2"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</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">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</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">h</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">j</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_5_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:34_1_1_2_5_1_2_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:34_1_1_2_5_1_2_1"/>
<font color="Maroon">j</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="txt" href="integra4.html#E6:34_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:34_1_1_2_5_1_2_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E5:34_1_1_2_5_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:34_1_1_2_5_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:34_1_1_2_5_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h</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">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra4.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:34_1_1_2_5_1_2_1"/><b>then </b>
<font color="Maroon">r</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">h</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">j</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="E9:34_1_1_2_5_1_2_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</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">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">h</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:34_1_1_2_5_1_2_1"/><i><font color="Green">E142</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</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">j</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E11:34_1_1_2_5_1_2_1"/>
( <font color="Maroon">h</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">h</font> <a href="rfunct_1.html#R2">is_bounded_below_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="E12:34_1_1_2_5_1_2_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>
<a NAME="E13:34_1_1_2_5_1_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> is <a href="seq_4.html#V3">bounded</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="E14:34_1_1_2_5_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</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">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T5">RCOMP_1:5</a></i>;<br/>

<b>thus </b><a NAME="E15:34_1_1_2_5_1_2_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</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">h</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">j</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_1_1_2_5_1_2"/><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"><font color="Maroon">h</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">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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"><font color="Maroon">h</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">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1_2_5_1_2"/>
<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">h</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">h</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:34_1_1_2_5_1"/>
 <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">j</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/>
<b>hence </b><a NAME="E8:34_1_1_2_5_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="integra4.html#E8:34_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="integra4.html#E17:34_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1_2_5"/>
<font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E7:34_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:34_1_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">UV1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">UV</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">LV</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E35:34_1_1_2"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_2_6"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_2_6_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E2:34_1_1_2_6_1"/>
<font color="Maroon">j</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">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E8:34_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:34_1_1_2_6_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">j</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="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:34_1_1_2_6_1"/><b>then </b>
( <font color="Maroon">UV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</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">j</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">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">LV</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <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">D</font>,<font color="Maroon">j</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">j</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E147</font></i>: <font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<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="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E8:34_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></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">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E6:34_1_1_2_6_1"/><i><font color="Green">E148</font></i>: 
<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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">D</font>,<font color="Maroon">j</font></span>)</span></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="34_1_1_2_6_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_2_6_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">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</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">j</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_6_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:34_1_1_2_6_1_2_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:34_1_1_2_6_1_2_1"/>
<font color="Maroon">j</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="txt" href="integra4.html#E17:34_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:34_1_1_2_6_1_2_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E5:34_1_1_2_6_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:34_1_1_2_6_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:34_1_1_2_6_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</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">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:34_1_1_2_6_1_2_1"/><b>then </b>
<font color="Maroon">r</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">j</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="E9:34_1_1_2_6_1_2_1"/><b>then </b><i><font color="Green">E150</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">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:34_1_1_2_6_1_2_1"/><i><font color="Green">E151</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">j</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/>
<a NAME="E11:34_1_1_2_6_1_2_1"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_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="E12:34_1_1_2_6_1_2_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>
<a NAME="E13:34_1_1_2_6_1_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V3">bounded</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="E14:34_1_1_2_6_1_2_1"/>
 <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">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="integra4.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T5">RCOMP_1:5</a></i>;<br/>

<b>thus </b><a NAME="E15:34_1_1_2_6_1_2_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</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">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra4.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_1_1_2_6_1_2"/><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"><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">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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"><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">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1_2_6_1_2"/>
<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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:34_1_1_2_6_1"/>
 <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">j</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/>
<b>hence </b><a NAME="E8:34_1_1_2_6_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="integra4.html#T3" target="_self">Th3</a>, , <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1_2_6"/>
<font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E15:34_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>

<a NAME="E3:34_1_1_2_6"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_1_2_6_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <font color="Maroon">G</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E2:34_1_1_2_6_2"/>
<font color="Maroon">j</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">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E8:34_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:34_1_1_2_6_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">j</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="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:34_1_1_2_6_2"/><b>then </b>
( <font color="Maroon">UV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</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">g</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">j</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">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">LV1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <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">g</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">j</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">j</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E154</font></i>: <font color="Maroon">G</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<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="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra4.html#E8:34_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span></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">j</font></span>)</span></span>)</span>

;<br/>
<a NAME="E6:34_1_1_2_6_2"/><i><font color="Green">E155</font></i>: 
<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">g</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">g</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">j</font></span>)</span></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="34_1_1_2_6_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_2_6_2_2"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</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">g</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">j</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_6_2_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:34_1_1_2_6_2_2_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E3:34_1_1_2_6_2_2_1"/>
<font color="Maroon">j</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="txt" href="integra4.html#E17:34_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:34_1_1_2_6_2_2_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E5:34_1_1_2_6_2_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:34_1_1_2_6_2_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:34_1_1_2_6_2_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</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">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:34_1_1_2_6_2_2_1"/><b>then </b>
<font color="Maroon">r</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">g</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">j</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="E9:34_1_1_2_6_2_2_1"/><b>then </b><i><font color="Green">E157</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</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">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">g</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">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:34_1_1_2_6_2_2_1"/><i><font color="Green">E158</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">j</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E11:34_1_1_2_6_2_2_1"/>
( <font color="Maroon">g</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="rfunct_1.html#R2">is_bounded_below_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="E12:34_1_1_2_6_2_2_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/>
<a NAME="E13:34_1_1_2_6_2_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> is <a href="seq_4.html#V3">bounded</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="E14:34_1_1_2_6_2_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</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">j</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="integra4.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T5">RCOMP_1:5</a></i>;<br/>

<b>thus </b><a NAME="E15:34_1_1_2_6_2_2_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">r</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">g</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">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra4.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_1_1_2_6_2_2"/><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"><font color="Maroon">g</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">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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"><font color="Maroon">g</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">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24">SEQ_4:24</a></i>;<br/>
<b>hence </b><a NAME="E3:34_1_1_2_6_2_2"/>
<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">g</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">g</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">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:34_1_1_2_6_2"/>
 <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">j</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/>
<b>hence </b><a NAME="E8:34_1_1_2_6_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="integra4.html#T3" target="_self">Th3</a>, , <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34_1_1_2_6"/>
<font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E3:34_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:34_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E37:34_1_1_2"/><b>then </b><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T24">ABSVALUE:24</a></i>;<br/>
<a NAME="E38:34_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E6:34_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E39:34_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E8:34_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E40:34_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E9:34_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E41:34_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 
;<br/>
<a NAME="E42:34_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E43:34_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra4.html#E1:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E44:34_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">osc1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="integra4.html#E1:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E45:34_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="integra4.html#E10:34_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="integra4.html#E3:34_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:34_1_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">osc2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:34_1"/><b>then </b>
<font color="Maroon">osc2</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E17:34_1"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K10">-</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra4.html#E4:34_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E18:34_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3">upper_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">h</font>,<font color="Maroon">T</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="integra4.html#E7:34"><i><font color="Green">E44</font></i></a>, <a class="ref" href="seq_2.html#T26">SEQ_2:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:34"/>
<font color="Maroon">h</font> <a href="integra1.html#R3">is_integrable_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


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