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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:102"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:102"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:102"/><i><font color="Green">E38</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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> =  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<a NAME="E5:102"/><i><font color="Green">E39</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>  <a href="relset_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="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:102_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_1"><b>suppose </b></a><a NAME="E1:102_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:102_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:102_1_1"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><b>hence </b><a NAME="E4:102_1_1"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="102_1_2"><b>suppose </b></a><a NAME="E1:102_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><a NAME="E2:102_1_2"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">A</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:102_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:102_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/>

<a NAME="E2:102_1_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E13:102_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:102_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E5:102_1_2"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:102_1_2_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 ;<br/>

<a NAME="E2:102_1_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E3:102_1_2_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:102_1_2_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc3.html#T4">MESFUNC3:4</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><a NAME="E8:102_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:102_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/><i><font color="Green">E94</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> = 
 <a href="finseq_1.html#K2">Seg</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/><a NAME="E10:102_1_2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/>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> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:102_1_2_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:102_1_2_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:102_1_2_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E4:102_1_2_4"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:102_1_2_4"/><b>then </b>
<font color="Maroon">x</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="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E6:102_1_2_4"/><b>then </b>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:102_1_2_4"><i><font color="Green">E98</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:102_1_2_4"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E16:102_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:102_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">F</font>,<font color="Maroon">a</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><b>set </b><font color="Maroon">x</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><b>reconsider </b><font color="Maroon">x</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/><i><font color="Green">E101</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font> = 
 <a href="finseq_1.html#K2">Seg</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/><a NAME="E14:102_1_2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:102_1_2_6"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:102_1_2_6"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E3:102_1_2_6"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:102_1_2_6"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E16:102_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E17:102_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E6:102_1_2_6"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E17:102_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:102_1_2"/><b>then </b>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E11:102_1_2"><i><font color="Green">E100</font></i></a>, , , , <a class="ref" href="mesfunc4.html#T3">MESFUNC4:3</a></i>;<br/><a NAME="E16:102_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E11:102_1_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="mesfunc5.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E17:102_1_2"/><i><font color="Green">E107</font></i>: 
1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E18:102_1_2"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:102_1_2_7"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:102_1_2_7"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E3:102_1_2_7"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:102_1_2_7"/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:102_1_2"/><b>then </b>
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> 1</span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E17:102_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="mesfunc3.html#T18">MESFUNC3:18</a></i>;<br/><b>hence </b><a NAME="E20:102_1_2"/>
 <a href="mesfunc5.html#K5" target="_self">integral'</a> <font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E16:102_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
