<?xml version="1.0"?>
<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/><b>let </b><font color="Maroon">A</font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_2.html#K4" target="_self">.|.</a> <font color="Maroon">B</font></span>)</span> <a href="prob_1.html#K10">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_2.html#K4" target="_self">.|.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="prob_1.html#K10">.</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:45"/><i><font color="Green">E41</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:45"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E7:45"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_2.html#K4" target="_self">.|.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="prob_1.html#K10">.</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#T68">PROB_1:68</a></i>;<br/>
<a NAME="E11:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<a NAME="E12:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<b>then </b><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">A</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E14:45"/>
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div>
