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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">P</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Olive" title="b1">Sigma</font><br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font>, <font color="Olive" title="b5">C</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Olive" title="b1">Sigma</font>  st <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Olive" title="b2">P</font> holds <br/><font color="Olive" title="b3">A</font>,<font color="Olive" title="b5">C</font>,<font color="Olive" title="b4">B</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Olive" title="b2">P</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font>, <font color="Olive" title="b4">C</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">C</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Olive" title="b1">P</font> holds <br/><font color="Olive" title="b2">A</font>,<font color="Olive" title="b4">C</font>,<font color="Olive" title="b3">B</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Olive" title="b1">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font> holds <br/><font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">C</font>,<font color="Olive" title="b2">B</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font>, <font color="Maroon" title="c6">C</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font> implies <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c5">B</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font> )</span><br/>











<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c5">B</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font></span><br/>

<a NAME="E2:40"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">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="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">B</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#T32" target="_self" title="PROB_2:th.32">Th32</a></span>;<br/>
<a NAME="E3:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c6">C</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">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="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">C</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>;<br/>
<a NAME="E4:40"/>
<font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c6">C</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="prob_2.html#T32" target="_self" title="PROB_2:th.32">Th32</a></span>;<br/>
<a NAME="E5:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c5">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#T31" target="_self" title="PROB_2:th.31">Th31</a></span>;<br/>
<a NAME="E6:40"/>
( <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="prob_2.html#T32" target="_self" title="PROB_2:th.32">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:40"/>
<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c5">B</font> <a href="prob_2.html#R2" title="PROB_2:pred.2">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="prob_2.html#T32" target="_self" title="PROB_2:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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