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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M1" title="MEASURE1:mode.1">Measure</a> of <font color="Maroon" title="c2">S</font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">F</font> being  <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F</font></span>)</span>
 ;<br/>

<a NAME="E2:19"/><i><font color="Green" title="E13">A2</font></i>: 
for <font color="Olive" title="b1">F</font> being  <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of <font color="Maroon" title="c2">S</font> holds   <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Olive" title="b1">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" title="c4">F</font> be   <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of <font color="Maroon" title="c2">S</font>;<br/>

<a NAME="E1:19_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="measure3.html#E1:19"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<a NAME="E2:19_1"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="measure3.html#T15" target="_self" title="MEASURE3:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure3.html#E1:19_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/>
( <font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 
<b>by </b><i><a class="ref" href="measure1.html#D5" title="MEASURE1:def.5">MEASURE1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon" title="c3">M</font> is    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="measure3.html#E2:19"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>


</div>
