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

<b>let </b><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">Omega</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Sigma</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</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="c1">I</font>,<font color="Olive" title="b1">Sigma</font> holds   <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c1">I</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font></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="c2">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">F</font> being   <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">Sigma</font> holds   <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c1">I</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font></span><br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c1">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">  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">I</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font></span><br/>







<b>set </b><font color="Maroon" title="c5">T</font> =  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">I</font>;<br/>
<a NAME="E1:30"/><i><font color="Green" title="E24">A0</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <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="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="c4">F</font>,<font color="Maroon" title="c1">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</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">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="c4">F</font>,<font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="c4">F</font>,<font color="Maroon" title="c1">I</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E25">Z</font></i>: 
<font color="Maroon" title="c6">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="c4">F</font>,<font color="Maroon" title="c1">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="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="c4">F</font>,<font color="Maroon" title="c1">I</font></span><br/>

<a NAME="E2:30_1"/>
for <font color="Olive" title="b1">S</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">S</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="c7">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green" title="E26">C0</font></i>: 
<font color="Maroon" title="c7">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">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="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font></span><br/>

<a NAME="E2:30_1_1"/><b>then </b><i><font color="Green" title="E27">C1</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="#E1:30_1"><i><font color="Green" title="E25">Z</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">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="c1">I</font><b> such that </b><br/><a NAME="E4:30_1_1"/><i><font color="Green" title="E28">C2</font></i>: 
<font color="Maroon" title="c7">S</font> <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="c4">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:30_1_1"><i><font color="Green" title="E26">C0</font></i></a>, <a class="ref" href="kolmog01.html#D7" target="_self" title="KOLMOG01:def.7">Def7</a></i>;<br/>
<a NAME="E5:30_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of  <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="c4">F</font> <a href="kolmog01.html#K2" title="KOLMOG01:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#T50" title="PROB_1:th.50">PROB_1:50</a>, <a class="txt" href="#E2:30_1_1"><i><font color="Green" title="E27">C1</font></i></a>, <a class="txt" href="#E4:30_1_1"><i><font color="Green" title="E28">C2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:30_1_1"/>
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="#E4:30_1_1"><i><font color="Green" title="E28">C2</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="E3:30_1"/>
<font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="c4">F</font>,<font color="Maroon" title="c1">I</font>
 <b>by </b><i><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="E2:30"/>
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">Omega</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">I</font> holds <br/> <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Olive" title="b1">A1</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="c4">F</font>,<font color="Maroon" title="c1">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">I</font> implies  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c6">A1</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="c4">F</font>,<font color="Maroon" title="c1">I</font> )</span><br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green" title="E25">B0</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c6">A1</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="c4">F</font>,<font color="Maroon" title="c1">I</font></span><br/>

<a NAME="E2:30_2"/><i><font color="Green" title="E26">B1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">S</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">S</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font></span><br/><b>let </b><font color="Maroon" title="c8">S</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="c8">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font> )</span><br/>



<b>assume </b><a NAME="E1:30_2_1"/><i><font color="Green" title="E27">C0</font></i>: 
<font color="Maroon" title="c8">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">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="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font></span><br/>

<a NAME="E2:30_2_1"/>
<font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</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">A1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></i>;<br/>
<b>hence </b><a NAME="E3:30_2_1"/>
<font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 <b>by </b><i><a class="txt" href="#E1:30_2_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a>, <a class="txt" href="#E1:30_2"><i><font color="Green" title="E25">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>
<a NAME="E3:30_2"/>
for <font color="Olive" title="b1">S</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">S</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="c7">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> )</span><br/>

<b>assume </b><a NAME="E1:30_2_2"/><i><font color="Green" title="E27">C0</font></i>: 
<font color="Maroon" title="c7">S</font> <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="c4">F</font>,<font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">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="c1">I</font><b> such that </b><br/><a NAME="E3:30_2_2"/><i><font color="Green" title="E28">C1</font></i>: 
<font color="Maroon" title="c7">S</font> <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="c4">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
 <b>by </b><i><a class="ref" href="kolmog01.html#D7" target="_self" title="KOLMOG01:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">S</font> = <font color="Maroon" title="c7">S</font> as   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font> <b>by </b><i><a class="txt" href="#E3:30_2_2"><i><font color="Green" title="E28">C1</font></i></a></i>;<br/>
<a NAME="E5:30_2_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c10">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:30_2_2_1"/>
<font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font>
 
<b>by </b><i><a class="txt" href="#E1:30_2_2"><i><font color="Green" title="E27">C0</font></i></a>, <a class="txt" href="#E2:30_2"><i><font color="Green" title="E26">B1</font></i></a></i>;<br/>
<a NAME="E3:30_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c9">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T50" title="PROB_1:th.50">PROB_1:50</a></i>;<br/>
<a NAME="E4:30_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font>
 
;<br/>
<b>hence </b><a NAME="E5:30_2_2_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D4" title="PROB_1:def.4">PROB_1: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="E6:30_2_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">S</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">CA1</font> =  <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font> as    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c9">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/>
<a NAME="E8:30_2_2"/>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c10">CA1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46" title="PROB_1:th.46">PROB_1:46</a></i>;<br/>
<a NAME="E9:30_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c9">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T50" title="PROB_1:th.50">PROB_1:50</a></i>;<br/>
<b>hence </b><a NAME="E10:30_2_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <font color="Maroon" title="c6">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</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>
<b>hence </b><a NAME="E4:30_2"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c6">A1</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="c4">F</font>,<font color="Maroon" title="c1">I</font>
 <b>by </b><i><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>
<b>hence </b><a NAME="E3:30"/>
 <a href="kolmog01.html#K6" title="KOLMOG01:func.6">tailSigmaField</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">I</font> is   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font>
 <b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E24">A0</font></i></a>, <a class="ref" href="prob_1.html#T32" title="PROB_1:th.32">PROB_1:32</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>
