<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">Omega</font>, <font color="Maroon" title="c2">I</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> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">Sigma</font><br/> for <font color="Olive" title="b4">F</font> being   <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c2">I</font>,<font color="Olive" title="b1">Sigma</font>  st <font color="Olive" title="b4">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Olive" title="b4">F</font>,<font color="Maroon" title="c2">I</font> &amp; not <font color="Olive" title="b2">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b2">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><b>let </b><font color="Maroon" title="c3">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="c3">Sigma</font><br/> for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Sigma</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b3">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Olive" title="b3">F</font>,<font color="Maroon" title="c2">I</font> &amp; not <font color="Olive" title="b1">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c3">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> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">Sigma</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b2">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c2">I</font> &amp; not <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">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">F</font> being   <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b1">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">I</font> &amp; not <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><b>let </b><font color="Maroon" title="c6">F</font> be    <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">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">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; not <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>











<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E24">A0</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</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="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</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">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<b>set </b><font color="Maroon" title="c7">Ie</font> = <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span>;<br/>
<a NAME="E3:31"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E27">Z</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:31_1"/>
 <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K5" title="KOLMOG01:func.5">futSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="kolmog01.html#D7" target="_self" title="KOLMOG01:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:31_1"><i><font color="Green" title="E27">Z</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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>
<a NAME="E4:31"/><i><font color="Green" title="E27">A3</font></i>: 
for <font color="Olive" title="b1">E</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> holds <br/><font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">E</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</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>, <font color="Olive" title="b2">q</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">E</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> holds <br/><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/><b>let </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">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="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> implies <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font> )</span><br/>





<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E28">B0</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/>

<a NAME="E2:31_2"/>
for <font color="Olive" title="b1">E</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font> holds  <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">E</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">E</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">E</font></span>)</span></span><br/>

<a NAME="E1:31_2_1"/>
 <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">E</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K5" title="KOLMOG01:func.5">futSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="kolmog01.html#D7" target="_self" title="KOLMOG01:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E2:31_2_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:31_2"><i><font color="Green" title="E28">B0</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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>
<a NAME="E3:31_2"/><b>then </b><i><font color="Green" title="E29">B1</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_2"/>
( ( <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or  not <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_1"><b>suppose </b></a><a NAME="E1:31_2_2_1"/><i><font color="Green" title="E30">C0</font></i>: 
( <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c11">E</font> = <font color="Maroon" title="c8">E</font> as  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">I</font> ;<br/><b>reconsider </b><font color="Maroon" title="c12">IE</font> = <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c11">E</font> as  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">I</font> <b>by </b><i><a class="txt" href="#E1:31_2_2_1"><i><font color="Green" title="E30">C0</font></i></a></i>;<br/><a NAME="E4:31_2_2_1"/>
<font color="Maroon" title="c11">E</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c12">IE</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E5:31_2_2_1"/><b>then </b>
<font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E24">A0</font></i></a>, <a class="txt" href="#E1:31_2"><i><font color="Green" title="E28">B0</font></i></a>, <a class="txt" href="#E3:31_2"><i><font color="Green" title="E29">B1</font></i></a>, <a class="ref" href="kolmog01.html#T14" target="_self" title="KOLMOG01:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E6:31_2_2_1"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="prob_2.html#D5" title="PROB_2:def.5">PROB_2:def 5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_2"><b>suppose </b></a><a NAME="E1:31_2_2_2"/><i><font color="Green" title="E30">C1</font></i>: 
(  not <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c11">e</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:31_2_2_2"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><a NAME="E4:31_2_2_2"/><b>then </b><i><font color="Green" title="E31">D2</font></i>: 
 <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">Omega</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="kolmog01.html#T3" target="_self" title="KOLMOG01:th.3">Th1</a></i>;<br/><a NAME="E5:31_2_2_2"/><i><font color="Green" title="E32">D3</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">Omega</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">u</font>, <font color="Maroon" title="c13">v</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">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="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">Omega</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font> )</span><br/>



<b>assume </b><a NAME="E1:31_2_2_2_1"/><i><font color="Green" title="E33">E0</font></i>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c1">Omega</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_2_2_1_1"/>
( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:31_2_2_2_1_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/><div class="add"><b>hence </b><a NAME="E2:31_2_2_2_1_1_1"/>
<font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="prob_2.html#T31" title="PROB_2:th.31">PROB_2:31</a>, <a class="ref" href="prob_2.html#T35" title="PROB_2:th.35">PROB_2:35</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:31_2_2_2_1_1_2"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/><div class="add"><a NAME="E2:31_2_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c3">Sigma</font>
 <b>by </b><i><a class="txt" href="#E1:31_2_2_2_1"><i><font color="Green" title="E33">E0</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:31_2_2_2_1_1_2"/>
<font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c13">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="prob_2.html#T31" title="PROB_2:th.31">PROB_2:31</a>, <a class="ref" href="prob_2.html#T36" title="PROB_2:th.36">PROB_2:36</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:31_2_2_2"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_2_2_2_1"/>
( <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:31_2_2_2_2_1_1"/>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/><div class="add"><b>hence </b><a NAME="E2:31_2_2_2_2_1_1"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="#E4:31_2_2_2"><i><font color="Green" title="E31">D2</font></i></a>, <a class="txt" href="#E5:31_2_2_2"><i><font color="Green" title="E32">D3</font></i></a>, <a class="txt" href="#E1:31_2"><i><font color="Green" title="E28">B0</font></i></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:31_2_2_2_2_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/><div class="add"><b>hence </b><a NAME="E2:31_2_2_2_2_1_2"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="prob_2.html#T31" title="PROB_2:th.31">PROB_2:31</a>, <a class="txt" href="#E4:31_2_2_2"><i><font color="Green" title="E31">D2</font></i></a>, <a class="txt" href="#E5:31_2_2_2"><i><font color="Green" title="E32">D3</font></i></a>, <a class="txt" href="#E3:31_2"><i><font color="Green" title="E29">B1</font></i></a>, <a class="txt" href="#E1:31_2_2_2"><i><font color="Green" title="E30">C1</font></i></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></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E7:31_2_2_2"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 ; <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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:31"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> holds <br/><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">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="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> implies <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font> )</span><br/>



<b>assume </b><a NAME="E1:31_3"/><i><font color="Green" title="E29">B0</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">E</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E3:31_3"/><i><font color="Green" title="E30">B1</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c10">E</font>
 <b>by </b><i><a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</a></i>;<br/>
<b>thus </b><a NAME="E4:31_3"/>
<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">q</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="#E1:31_3"><i><font color="Green" title="E29">B0</font></i></a>, <a class="txt" href="#E3:31_3"><i><font color="Green" title="E30">B1</font></i></a>, <a class="txt" href="#E4:31"><i><font color="Green" title="E27">A3</font></i></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>reconsider </b><font color="Maroon" title="c8">T</font> =  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> as   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font> <b>by </b><i><a class="ref" href="kolmog01.html#T15" target="_self" title="KOLMOG01:th.15">Th16</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green" title="E29">a5</font></i>: 
<font color="Maroon" title="c8">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <font color="Maroon" title="c8">T</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green" title="E30">A6</font></i>: 
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_4"/>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_4_1"/>
 <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:31_4_1_1"/>
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c10">E</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E3:31_4_1_1"/><i><font color="Green" title="E31">E0</font></i>: 
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c10">E</font>
 <b>by </b><i><a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</a></i>;<br/>
<a NAME="E4:31_4_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c10">E</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T104" title="RELAT_1:th.104">RELAT_1:104</a></i>;<br/>
<a NAME="E5:31_4_1_1"/><b>then </b><i><font color="Green" title="E32">E2</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c10">E</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/>
<a NAME="E6:31_4_1_1"/>
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<a NAME="E7:31_4_1_1"/><b>then </b>
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c10">E</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:31_4_1_1"><i><font color="Green" title="E32">E2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:31_4_1_1"/><b>then </b>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c10">E</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E9:31_4_1_1"/>
<font color="Maroon" title="c9">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:31_4_1_1"><i><font color="Green" title="E31">E0</font></i></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="E2:31_4_1"/>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</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>

<a NAME="E2:31_4"/>
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:31_4_2"/><i><font color="Green" title="E31">E0</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span></span><br/>

<a NAME="E2:31_4_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span>
 
;<br/>
<a NAME="E3:31_4_2"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:31_4_2"><i><font color="Green" title="E31">E0</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:31_4_2"/><i><font color="Green" title="E32">E1</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:31_4_2"/><i><font color="Green" title="E33">E2</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="#E5:31_4_2"><i><font color="Green" title="E32">E1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Olive" title="b1">i</font> );<br/>
<b>consider </b><font color="Maroon" title="c12">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:31_4_2"/><i><font color="Green" title="E34">E3</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> iff ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E10:31_4_2"/><i><font color="Green" title="E35">E4</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_4_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4_2_1_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E36">D0</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E7:31_4_2"><i><font color="Green" title="E33">E2</font></i></a>, <a class="ref" href="zfmisc_1.html#T52" title="ZFMISC_1:th.52">ZFMISC_1:52</a></i>
;<br/>
<a NAME="E2:31_4_2_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a></i>;<br/>
<a NAME="E3:31_4_2_1_1"/><b>then </b><i><font color="Green" title="E37">D1</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/>
<a NAME="E4:31_4_2_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:31_4_2_1_1"><i><font color="Green" title="E36">D0</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:31_4_2_1_1"/><b>then </b><i><font color="Green" title="E38">D2</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c11">i</font>
 
<b>by </b><i><a class="txt" href="#E3:31_4_2_1_1"><i><font color="Green" title="E37">D1</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E6:31_4_2_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="#E7:31_4_2"><i><font color="Green" title="E33">E2</font></i></a></i>;<br/>
<a NAME="E7:31_4_2_1_1"/><b>then </b>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">E</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">Fi</font> = <font color="Maroon" title="c6">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c11">i</font> as   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font> <b>by </b><i><a class="txt" href="#E7:31_4_2"><i><font color="Green" title="E33">E2</font></i></a>, <a class="ref" href="kolmog01.html#D2" target="_self" title="KOLMOG01:def.2">Def2</a></i>;<br/>
<a NAME="E10:31_4_2_1_1"/><i><font color="Green" title="E39">D3</font></i>: 
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <font color="Maroon" title="c14">Fi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">Fi</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<a NAME="E11:31_4_2_1_1"/>
<font color="Maroon" title="c14">Fi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <font color="Maroon" title="c14">Fi</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<a NAME="E12:31_4_2_1_1"/><b>then </b><i><font color="Green" title="E40">D4</font></i>: 
 <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c13">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c11">i</font>
 
<b>by </b><i><a class="txt" href="#E5:31_4_2_1_1"><i><font color="Green" title="E38">D2</font></i></a>, <a class="txt" href="#E10:31_4_2_1_1"><i><font color="Green" title="E39">D3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font> <b>by </b><i><a class="txt" href="#E5:31_4_2"><i><font color="Green" title="E32">E1</font></i></a>, <a class="txt" href="#E7:31_4_2"><i><font color="Green" title="E33">E2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E14:31_4_2_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="#E12:31_4_2_1_1"><i><font color="Green" title="E40">D4</font></i></a>, <a class="txt" href="#E7:31_4_2"><i><font color="Green" title="E33">E2</font></i></a>, <a class="txt" href="#E5:31_4_2"><i><font color="Green" title="E32">E1</font></i></a>, <a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</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="E2:31_4_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 <b>by </b><i><a class="txt" href="#E9:31_4_2"><i><font color="Green" title="E34">E3</font></i></a>, <a class="txt" href="#E7:31_4_2"><i><font color="Green" title="E33">E2</font></i></a>, <a class="txt" href="#E5:31_4_2"><i><font color="Green" title="E32">E1</font></i></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>
<a NAME="E11:31_4_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="#E9:31_4_2"><i><font color="Green" title="E34">E3</font></i></a></i>;<br/>
<a NAME="E12:31_4_2"/><b>then </b><i><font color="Green" title="E36">E5</font></i>: 
<font color="Maroon" title="c12">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:31_4_2"/>
 <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E14:31_4_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:31_4_2"><i><font color="Green" title="E35">E4</font></i></a>, <a class="txt" href="#E12:31_4_2"><i><font color="Green" title="E36">E5</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="E3:31_4"/>
 <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</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>
<a NAME="E9:31"/><i><font color="Green" title="E31">A7</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">Sigma</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> holds <br/><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">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="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> implies <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font> )</span><br/>



<b>assume </b><a NAME="E1:31_5"/><i><font color="Green" title="E32">Z</font></i>: 
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font></span><br/>

<a NAME="E2:31_5"/><i><font color="Green" title="E33">C0</font></i>: 
not  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c11">E</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font>;<br/>
<a NAME="E2:31_5_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c11">E</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T42" title="PROB_1:th.42">PROB_1:42</a></i>;<br/>
<b>hence </b><a NAME="E3:31_5_1"/>
not  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</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>
<a NAME="E3:31_5"/><i><font color="Green" title="E34">c0</font></i>: 
 <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font> )</span><br/>

<b>assume </b><a NAME="E1:31_5_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/>

<b>then consider </b><font color="Maroon" title="c12">E</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E3:31_5_2"/><i><font color="Green" title="E35">D0</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c12">E</font>
 <b>by </b><i><a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</a></i>;<br/>
<a NAME="E4:31_5_2"/>
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c12">E</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c12">E</font></span>)</span> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font> )</span><br/>

<b>assume </b><a NAME="E1:31_5_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c12">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/>

<b>then consider </b><font color="Maroon" title="c14">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_5_2_1"/><i><font color="Green" title="E36">F0</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c14">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c12">E</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>thus </b><a NAME="E4:31_5_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 <b>by </b><i><a class="txt" href="#E3:31_5_2_1"><i><font color="Green" title="E36">F0</font></i></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>
<a NAME="E5:31_5_2"/><b>then </b>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c12">E</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E6:31_5_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 <b>by </b><i><a class="txt" href="#E3:31_5_2"><i><font color="Green" title="E35">D0</font></i></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>
<a NAME="E4:31_5"/><i><font color="Green" title="E35">B2</font></i>: 
 <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">intersection_stable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_3"><b>proof </b></a><div class="add">

<a NAME="E1:31_5_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be    <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"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> implies <font color="Maroon" title="c11">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> )</span><br/>



<b>assume </b><a NAME="E1:31_5_3_1"/><i><font color="Green" title="E36">D0</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">E1</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E3:31_5_3_1"/><i><font color="Green" title="E37">D1</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c13">E1</font>
 <b>by </b><i><a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">E2</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E5:31_5_3_1"/><i><font color="Green" title="E38">D2</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c14">E2</font>
 <b>by </b><i><a class="txt" href="#E1:31_5_3_1"><i><font color="Green" title="E36">D0</font></i></a>, <a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</a></i>;<br/>
<a NAME="E6:31_5_3_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">E1</font>, <font color="Olive" title="b3">E2</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Olive" title="b2">E1</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Olive" title="b3">E2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">z</font> be    <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">E1</font>, <font color="Olive" title="b2">E2</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font>  st <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Olive" title="b1">E1</font> holds <br/><font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Olive" title="b2">E2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c16">E1</font>, <font color="Maroon" title="c17">E2</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c16">E1</font> implies <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c17">E2</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:31_5_3_1_1"/><i><font color="Green" title="E39">Z</font></i>: 
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c16">E1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c17">E2</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c18">E3</font> = <font color="Maroon" title="c16">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c17">E2</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">I</font> ;<br/>
<a NAME="E3:31_5_3_1_1"/><i><font color="Green" title="E40">F1</font></i>: 
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span> or <font color="Maroon" title="c19">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:31_5_3_1_1_1"/>
<font color="Maroon" title="c19">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c20">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_5_3_1_1_1"/><i><font color="Green" title="E41">G0</font></i>: 
( <font color="Maroon" title="c19">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font> &amp; <font color="Maroon" title="c20">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:31_5_3_1_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c17">E2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</a></i>;<br/>
<a NAME="E5:31_5_3_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c17">E2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></i>;<br/>
<a NAME="E6:31_5_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:31_5_3_1_1_1"><i><font color="Green" title="E41">G0</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:31_5_3_1_1_1"/>
<font color="Maroon" title="c19">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:31_5_3_1_1_1"><i><font color="Green" title="E41">G0</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
<a NAME="E4:31_5_3_1_1"/>
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<a NAME="E5:31_5_3_1_1"/><b>then </b>
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:31_5_3_1_1"><i><font color="Green" title="E40">F1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:31_5_3_1_1"/><b>then </b>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c16">E1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <font color="Maroon" title="c18">E3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E7:31_5_3_1_1"/>
<font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c17">E2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:31_5_3_1_1"><i><font color="Green" title="E39">Z</font></i></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>
<a NAME="E7:31_5_3_1"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c14">E2</font></span>)</span> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">E2</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:31_5_3_1"><i><font color="Green" title="E37">D1</font></i></a>, <a class="txt" href="#E5:31_5_3_1"><i><font color="Green" title="E38">D2</font></i></a></i>;<br/>
<a NAME="E8:31_5_3_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c14">E2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:31_5_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="ref" href="kolmog01.html#D10" target="_self" title="KOLMOG01:def.10">Def10</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="E2:31_5_3"/>
 <a href="kolmog01.html#K8" title="KOLMOG01:func.8">finSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">intersection_stable</a>
 <b>by </b><i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</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>
<a NAME="E5:31_5"/>
 <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font> )</span><br/>

<b>assume </b><a NAME="E1:31_5_4"/><i><font color="Green" title="E36">d0</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/>

<b>set </b><font color="Maroon" title="c12">Ie</font> = <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span>;<br/>
<a NAME="E2:31_5_4"/>
 <a href="kolmog01.html#K4" title="KOLMOG01:func.4">sigUn</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K5" title="KOLMOG01:func.5">futSigmaFields</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="kolmog01.html#D7" target="_self" title="KOLMOG01:def.7">Def7</a></i>;<br/>
<a NAME="E3:31_5_4"/><b>then </b><i><font color="Green" title="E37">D1</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:31_5_4"><i><font color="Green" title="E36">d0</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:31_5_4"/>
 <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font> )</span><br/>

<b>assume </b><a NAME="E1:31_5_4_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/>

<b>then consider </b><font color="Maroon" title="c14">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:31_5_4_1"/><i><font color="Green" title="E38">F0</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">S</font> &amp; <font color="Maroon" title="c14">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>thus </b><a NAME="E4:31_5_4_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 <b>by </b><i><a class="txt" href="#E3:31_5_4_1"><i><font color="Green" title="E38">F0</font></i></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>
<a NAME="E5:31_5_4"/><b>then </b>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="kolmog01.html#K3" title="KOLMOG01:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c2">I</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E6:31_5_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 <b>by </b><i><a class="txt" href="#E3:31_5_4"><i><font color="Green" title="E37">D1</font></i></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="E6:31_5"/>
<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="kolmog01.html#T10" target="_self" title="KOLMOG01:th.10">Th8</a>, <a class="txt" href="#E3:31_5"><i><font color="Green" title="E34">c0</font></i></a>, <a class="txt" href="#E2:31_5"><i><font color="Green" title="E33">C0</font></i></a>, <a class="txt" href="#E5:31"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E4:31_5"><i><font color="Green" title="E35">B2</font></i></a>, <a class="txt" href="#E1:31_5"><i><font color="Green" title="E32">Z</font></i></a>, <a class="txt" href="#E8:31"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="#E7:31"><i><font color="Green" title="E29">a5</font></i></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>
<a NAME="E10:31"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="#E2:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E9:31"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/>
<a NAME="E11:31"/><b>then </b>
<font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_2.html#D5" title="PROB_2:def.5">PROB_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:31"/>
( <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="kolmog01.html#T2" target="_self" title="KOLMOG01:th.2">Th15</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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