<?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>assume </b><a NAME="E1:46"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font> &amp;  <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/>

<a NAME="E2:46"/>
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><a class="txt" href="mesfunc5.html#E12:46"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">F</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>, <font color="Maroon">a</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:46"/><i><font color="Green">E37</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> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E12:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mesfunc3.html#T14">MESFUNC3:14</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></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">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E6:46"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E7:46"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</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="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> 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">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<a NAME="E8:46"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:46"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</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#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">sumx</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E11:46"/><i><font color="Green">E42</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> implies <font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E12:46"/><i><font color="Green">E43</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E12:46"><i><font color="Green">E43</font></i></a>, <a class="txt" href="mesfunc5.html#E13:46"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E13:46"/><i><font color="Green">E44</font></i>: 
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/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><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="E1:46_1"/>
<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/>

<a NAME="E2:46_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>assume </b><a NAME="E3:46_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:46_1"/><b>then </b>
 ex <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">v</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>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1"/>
contradiction
 <b>by </b><i><a class="ref" href="mesfunc5.html#T7" target="_self">Th7</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:46"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E15:46"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_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>;<br/>

<b>assume </b><a NAME="E1:46_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<b>assume </b><a NAME="E2:46_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</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="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:46_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:46_2"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:46_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:46_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:46_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T8" target="_self">Th8</a>, <a class="ref" href="mesfunc5.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E9:46_2"/><b>then </b>
<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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E10:46_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T8" target="_self">Th8</a>, <a class="ref" href="mesfunc5.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T8" target="_self">Th8</a>, , </i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>
;<br/>
<b>hence </b><a NAME="E12:46_2"/>
<font color="Maroon">sumx</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:46"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<a NAME="E17:46"/><b>then </b>
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E18:46"/>
( <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> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; ( 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> ) &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:46"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mesfunc5.html#T5" target="_self">Th5</a>, , </i>;<br/>


</div>
