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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> holds <br/>( <font color="Olive" title="b1">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> iff for <font color="Olive" title="b2">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b2">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> implies ( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> iff for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> iff for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<a NAME="E2:18"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> implies for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E2:18_1"/>
for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">T</font> be   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E2:18_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:18_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E1:18_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="#E2:18_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra4.html#T9" target="_self" title="INTEGRA4:th.9">Th9</a></i>;<br/>
<a NAME="E4:18_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E1:18_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="#E2:18_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra4.html#T8" target="_self" title="INTEGRA4:th.8">Th8</a></i>;<br/>
<a NAME="E5:18_1_1"/>
 <a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="integra1.html#D17" title="INTEGRA1:def.17">INTEGRA1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:18_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="#E4:18_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18_1"/>
for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:18"/>
( ( for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green" title="E20">A8</font></i>: 
for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a></span><br/>

<a NAME="E2:18_2"/><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V2" title="INTEGRA1:attr.2">upper_integrable</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V3" title="INTEGRA1:attr.3">lower_integrable</a> )
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="integra4.html#T10" target="_self" title="INTEGRA4:th.10">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">T</font> being   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:18_2"/><i><font color="Green" title="E22">A10</font></i>: 
(  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="integra4.html#T11" target="_self" title="INTEGRA4:th.11">Th11</a></i>;<br/>
<a NAME="E5:18_2"/>
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:18_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="#E4:18_2"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<a NAME="E6:18_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E4:18_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="integra4.html#T9" target="_self" title="INTEGRA4:th.9">Th9</a></i>;<br/>
<a NAME="E7:18_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K11" title="INTEGRA1:func.11">upper_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E4:18_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="integra4.html#T8" target="_self" title="INTEGRA4:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E8:18_2"/>
<font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>
 <b>by </b><i><a class="txt" href="#E2:18_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="integra1.html#D17" title="INTEGRA1:def.17">INTEGRA1:def 17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
( <font color="Maroon" title="c2">f</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> iff for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>  st  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">upper_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="#E2:18"><i><font color="Green" title="E19">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
