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<div class="add">

<b>let </b><font color="Maroon">Omega</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/>





<a NAME="E1:55"/>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</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="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</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="prob_4.html#K8" target="_self">COM</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> 0 holds <br/><font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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">Omega</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 ;<br/>

<b>assume </b><a NAME="E2:55_1"/><i><font color="Green">E36</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="prob_4.html#K8" target="_self">COM</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E3:55_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">Sigma</font> &amp;  ex <font color="Olive">C1</font> being   <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</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="55_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="E2:55_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">B2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C2</font> being   <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</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="txt" href="prob_4.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">C2</font> being    <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E4:55_1_1"/><i><font color="Green">E48</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="E5:55_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </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="E7:55_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="prob_4.html#E1:55_1"><i><font color="Green">E24</font></i></a></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">E51</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="E9:55_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="E10:55_1_1"/><b>then </b><i><font color="Green">E52</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="ref" href="prob_4.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:55_1_1"/><i><font color="Green">E54</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="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_2"/><i><font color="Green">E55</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">Sigma</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">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_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">Sigma</font> <b>by </b><i>, <a class="ref" href="prob_4.html#T3" target="_self">Th3</a>, <a class="ref" href="prob_1.html#T41">PROB_1:41</a></i>;<br/>
<a NAME="E2:55_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">O1</font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</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">Sigma</font> <b>by </b><i>, <a class="ref" href="prob_4.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:55_1_1_2_1"/>
( <font color="Maroon">B2</font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> &amp; <font color="Maroon">D2</font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> )
 
<b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<a NAME="E5:55_1_1_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</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> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="prob_1.html#T75">PROB_1:75</a>, , <a class="ref" href="prob_4.html#T3" target="_self">Th3</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="E6:55_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">Sigma</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">P</font> <a href="funct_1.html#K1">.</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> 0 )
 <b>by </b><i>, <a class="txt" href="prob_4.html#E7:55_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="prob_1.html#D13">PROB_1:def 13</a>, <a class="txt" href="prob_4.html#E5:55_1_1"><i><font color="Green">E49</font></i></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="E2:55_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="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="prob_4.html#E4:55_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="prob_4.html#E1:55_1"><i><font color="Green">E24</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E12:55_1_1"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C1</font> being   <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</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="ref" href="prob_1.html#T42">PROB_1:42</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:55_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:55"/>
 <a href="prob_4.html#K8" target="_self">COM</a> <font color="Maroon">P</font> <a href="prob_4.html#R1" target="_self">is_complete</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div>
