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<b>let </b><font color="Maroon" title="c1">Omega</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Sigma</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Olive" title="b1">Sigma</font><br/> for <font color="Olive" title="b4">P</font> being   <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Olive" title="b1">Sigma</font>  st <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Olive" title="b4">P</font> holds <br/><font color="Olive" title="b2">A</font>,<span class="p1">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Olive" title="b1">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Olive" title="b3">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Olive" title="b4">P</font></span><br/><b>let </b><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font><br/> for <font color="Olive" title="b3">P</font> being   <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Olive" title="b3">P</font> holds <br/><font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Olive" title="b2">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Olive" title="b3">P</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Olive" title="b1">P</font> holds <br/><font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c5">P</font> be    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c5">P</font> implies <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c5">P</font> )</span><br/>









<b>assume </b><a NAME="E1:31"/>
<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c5">P</font></span><br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_2.html#D5" target="_self" title="PROB_2:def.5">Def5</a></i>;<br/>
<font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T47" title="XBOOLE_1:th.47">XBOOLE_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:31"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="prob_1.html#T69" title="PROB_1:th.69">PROB_1:69</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_1.html#T68" title="PROB_1:th.68">PROB_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E4:31"/>
<font color="Maroon" title="c3">A</font>,<span class="p1">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="ref" href="prob_2.html#D5" target="_self" title="PROB_2:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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