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<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">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>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/>









<b>assume </b><a NAME="E1:35"/>
<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>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E37</font></i>: 
<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> <a href="hidden.html#R1">=</a> <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/>;<br/>
<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> <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> = 
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
<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#K8">\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<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#K8">\</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>
<b>by </b><i><a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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> 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> <font color="Maroon">A</font></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>
<b>by </b><i>, </i>
<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">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>

<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> <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><a class="ref" href="prob_1.html#T68">PROB_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E4:35"/>
<font color="Maroon">A</font>,<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> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


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