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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be    <a href="measure4.html#M1" target="_self">C_Measure</a> of <font color="Maroon">X</font>;<br/>



<b>reconsider </b><font color="Maroon">M</font> =  <a href="measure4.html#K3" target="_self">sigma_Meas</a> <font color="Maroon">C</font> as    <a href="measure1.html#M1">Measure</a> of  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E2:25"/>
for <font color="Olive">F</font> being  <a href="measure1.html#NM4">Sep_Sequence</a> of  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">C</font> holds  <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure4.html#K2" target="_self">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Olive">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Olive">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="measure1.html#NM4">Sep_Sequence</a> of  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:25_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="measure4.html#K2" target="_self">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">b</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="measure1.html#K1">.</a> <font color="Maroon">A</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E6:25_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F'</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<a NAME="E7:25_1"/>
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E3:25_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E5:25_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:25_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="measure4.html#E6:25_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E10:25_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure4.html#E2:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="measure4.html#E4:25_1"><i><font color="Green">E24</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E11:25_1"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure4.html#K2" target="_self">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E2:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="measure4.html#E4:25_1"><i><font color="Green">E24</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
 <a href="measure4.html#K3" target="_self">sigma_Meas</a> <font color="Maroon">C</font> is    <a href="measure1.html#M3">sigma_Measure</a> of  <a href="measure4.html#K1" target="_self">sigma_Field</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="measure3.html#T17">MEASURE3:17</a></i>;<br/>


</div>
