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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:44"/><i><font color="Green">E29</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:44"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_2.html#K4" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_2.html#K4" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="prob_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E3:44_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="prob_2.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:44"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="prob_2.html#R1" target="_self">are_independent_respect_to</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="prob_2.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E7:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E8:44"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_2.html#K4" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="prob_2.html#E1:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="prob_2.html#D7" target="_self">Def7</a></i>;<br/>


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