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<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#M1">Measure</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">F</font> being  <a href="measure1.html#NM4">Sep_Sequence</a> of <font color="Maroon">S</font> holds  <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">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>
 ;<br/>

<a NAME="E2:19"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">F</font> being  <a href="measure1.html#NM4">Sep_Sequence</a> of <font color="Maroon">S</font> holds   <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> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Olive">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_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 <font color="Maroon">S</font>;<br/>

<a NAME="E1:19_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">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/>;<br/>
<a NAME="E2:19_1"/>
 <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="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">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
 <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> <font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" target="_self">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/>
( <font color="Maroon">M</font> is <a href="supinf_2.html#V6">nonnegative</a> &amp; <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 
<b>by </b><i><a class="ref" href="measure1.html#D5">MEASURE1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon">M</font> is    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>


</div>
