<?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">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">T</font> be    <a href="measure2.html#M1" target="_self">N_Measure_fam</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> holds <br/><font color="Olive">A</font> is    <a href="measure1.html#M4">measure_zero</a> of <font color="Maroon">M</font>
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="measure1.html#K5">rng</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">G</font> = <font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>;<br/>
<a NAME="E4:16"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:16"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> 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="measure1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:16_1"/>
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="measure2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E2:16_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> is    <a href="measure1.html#M4">measure_zero</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_1"/><b>then </b>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="measure1.html#D13">MEASURE1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> implies <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="measure1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:16"/><b>then </b>
 <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="measure1.html#K1">.</a> 0
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T67">SUPINF_2:67</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
 <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> <a href="hidden.html#R1">=</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> <a href="measure1.html#K1">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
 <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> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" target="_self">union</a> <font color="Maroon">T</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="measure2.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E10:16"/>
<font color="Maroon">M</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" target="_self">union</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#D4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E12:16"/><b>then </b>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" target="_self">union</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<b>hence </b><a NAME="E13:16"/>
 <a href="measure2.html#K2" target="_self">union</a> <font color="Maroon">T</font> is    <a href="measure1.html#M4">measure_zero</a> of <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D13">MEASURE1:def 13</a></i>;<br/>


</div>
