<?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/>





<a NAME="E1:31"/>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><br/> for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K2" target="_self">COM</a> <font color="Maroon">S</font>,<font color="Maroon">M</font> &amp; <font color="Olive">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> &amp; <span class="p1">(<span class="default"><a href="measure3.html#K4" target="_self">COM</a> <font color="Maroon">M</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  holds <br/><font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K2" target="_self">COM</a> <font color="Maroon">S</font>,<font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K2" target="_self">COM</a> <font color="Maroon">S</font>,<font color="Maroon">M</font>
 ;<br/>

<b>assume </b><a NAME="E2:31_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><a href="measure3.html#K4" target="_self">COM</a> <font color="Maroon">M</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/>

<a NAME="E3:31_1"/>
 ex <font color="Olive">B1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C1</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">B2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:31_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C2</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C2</font> )
 <b>by </b><i><a class="ref" href="measure3.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>consider </b><font color="Maroon">C2</font> being    <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:31_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C2</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:31_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i>, , <a class="ref" href="measure3.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">D2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:31_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">C1</font> = <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span>;<br/>
<i><font color="Green">E40</font></i>: <font color="Maroon">A</font> = 
<font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C2</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<b>set </b><font color="Maroon">O</font> = <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font>;<br/>
<a NAME="E9:31_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="txt" href="measure3.html#E2:31_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finsub_1.html#D1">FINSUB_1:def 1</a></i>;<br/>
<a NAME="E10:31_1_1"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:31_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> )
 
<b>by </b><i><a class="txt" href="measure3.html#E2:31_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:31_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> is    <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_2"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">O</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">O</font> &amp; <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">O</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="31_1_1_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">O1</font> = <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="measure3.html#E4:31_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = <font color="Maroon">B2</font>, <font color="Maroon">D2</font> = <font color="Maroon">D2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="txt" href="measure3.html#E2:31_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E3:31_1_1_2_1"/>
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="measure1.html#K2">\/</a> <font color="Maroon">D2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure3.html#E2:31_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="measure3.html#E2:31_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="measure1.html#T64">MEASURE1:64</a></i>;<br/>
<a NAME="E4:31_1_1_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">M</font> <a href="measure1.html#K1">.</a> <font color="Maroon">O1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E5:31_1_1_2_1"/>
<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="E6:31_1_1_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <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> <font color="Maroon">O1</font>
 
<b>by </b><i><a class="ref" href="measure1.html#D4">MEASURE1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font>
;<br/>

<b>thus </b><a NAME="E7:31_1_1_2_1"/>
( <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">M</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">D2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure3.html#E5:31_1_1"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:31_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> is    <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="xboole_0.html#K1">{}</a> 
;<br/>

<b>thus </b><a NAME="E13:31_1_1"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C1</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <font color="Olive">C1</font> )
 <b>by </b><i>, <a class="txt" href="measure3.html#E7:31_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="measure3.html#K2" target="_self">COM</a> <font color="Maroon">S</font>,<font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31"/>
 <a href="measure3.html#K4" target="_self">COM</a> <font color="Maroon">M</font> <a href="measure3.html#R1" target="_self">is_complete</a>  <a href="measure3.html#K2" target="_self">COM</a> <font color="Maroon">S</font>,<font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div>
