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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">P</font> being   <a href="prob_1.html#M3" title="PROB_1:mode.3">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> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">Sigma</font>  st <font color="Olive" title="b3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Olive" title="b4">B</font>,<font color="Olive" title="b2">P</font> holds <br/><font color="Olive" title="b4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Olive" title="b3">A</font>,<font color="Olive" title="b2">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">P</font> being   <a href="prob_1.html#M3" title="PROB_1:mode.3">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> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Olive" title="b3">B</font>,<font color="Olive" title="b1">P</font> holds <br/><font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c3">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"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Olive" title="b2">B</font>,<font color="Maroon" title="c3">P</font> holds <br/><font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c3">P</font></span><br/><b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">P</font> implies <font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">P</font> )</span><br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E10">A0</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Maroon" title="c5">B</font>,<font color="Maroon" title="c3">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="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">P</font></span><br/>

<a NAME="E2:9"/>
for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">p</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">p</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">p</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"> ( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">p</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><br/>



<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E11">B0</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> or <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">p</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><br/>

<b>assume </b><a NAME="E2:9_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">p</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><br/>

<a NAME="E3:9_1"/><b>then </b>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E10">A0</font></i></a>, <a class="txt" href="#E1:9_1"><i><font color="Green" title="E11">B0</font></i></a>, <a class="ref" href="kolmog01.html#T7" target="_self" title="KOLMOG01:th.7">Th5</a></i>;<br/>
<b>hence </b><a NAME="E4:9_1"/>
<font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c7">p</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="ref" href="prob_2.html#T31" title="PROB_2:th.31">PROB_2:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K1" title="KOLMOG01:func.1">Indep</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="ref" href="kolmog01.html#T7" target="_self" title="KOLMOG01:th.7">Th5</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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