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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">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>let </b><font color="Maroon">C</font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font>;<br/>











<b>thus </b><a NAME="E1:30"/>
( <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="prob_2.html#R2" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font> implies ( <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><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> <a href="prob_1.html#K6">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</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> <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">C</font></span>)</span> &amp; <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> &amp; <font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="prob_2.html#R2" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b>
( <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><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> <a href="prob_1.html#K6">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</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> <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">C</font></span>)</span> &amp; <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> &amp; <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">C</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">C</font></span>)</span> &amp; <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">C</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">B</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">C</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
( <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><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> <a href="prob_1.html#K6">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</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> <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">C</font></span>)</span> &amp; <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> &amp; <font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font>,<font color="Maroon">C</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><b>end;</b></div>

<b>assume </b><a NAME="E2:30"/>
( <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><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> <a href="prob_1.html#K6">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</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> <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">C</font></span>)</span> &amp; <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> &amp; <font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E3:30"/><b>then </b>
( <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><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> <a href="prob_1.html#K6">/\</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</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> <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">C</font></span>)</span> &amp; <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> &amp; <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">C</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">C</font></span>)</span> &amp; <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">C</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">B</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">C</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:30"/>
<font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="prob_2.html#R2" target="_self">are_independent_respect_to</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div>
