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

<b>let </b><font color="Maroon" title="c5">X1</font>, <font color="Maroon" title="c6">X2</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</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">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font> holds <br/> ( <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> iff  ex <font color="Olive" title="b2">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> st <font color="Olive" title="b1">S</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="c4">F</font>,<font color="Olive" title="b2">E</font> ) ) &amp; ( for <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font> holds <br/> ( <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font> iff  ex <font color="Olive" title="b2">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> st <font color="Olive" title="b1">S</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="c4">F</font>,<font color="Olive" title="b2">E</font> ) ) implies <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X2</font> )</span><br/>

<b>assume </b><a NAME="E1:29_1_2"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font> holds <br/> ( <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> iff  ex <font color="Olive" title="b2">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> st <font color="Olive" title="b1">S</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="c4">F</font>,<font color="Olive" title="b2">E</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font> st <br/>( ( <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font> implies  ex <font color="Olive" title="b2">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> st <font color="Olive" title="b1">S</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="c4">F</font>,<font color="Olive" title="b2">E</font> ) implies (  ex <font color="Olive" title="b2">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> st <font color="Olive" title="b1">S</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="c4">F</font>,<font color="Olive" title="b2">E</font> &amp; not <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font> ) ) or <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X2</font> )</span><br/>

<b>assume </b><a NAME="E2:29_1_2"/><i><font color="Green" title="E23">A2</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font> holds <br/> ( <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font> iff  ex <font color="Olive" title="b2">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> st <font color="Olive" title="b1">S</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="c4">F</font>,<font color="Olive" title="b2">E</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">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">S</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="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> iff <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font> )</span><br/><a NAME="E1:29_1_2_1"/>
( ( <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> implies  ex <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="c1">I</font> st <font color="Maroon" title="c7">S</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="c4">F</font>,<font color="Olive" title="b1">E</font> ) &amp; (  ex <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="c1">I</font> st <font color="Maroon" title="c7">S</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="c4">F</font>,<font color="Olive" title="b1">E</font> implies <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> ) &amp; ( <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font> implies  ex <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="c1">I</font> st <font color="Maroon" title="c7">S</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="c4">F</font>,<font color="Olive" title="b1">E</font> ) &amp; (  ex <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="c1">I</font> st <font color="Maroon" title="c7">S</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="c4">F</font>,<font color="Olive" title="b1">E</font> implies <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font> ) )
 <b>by </b><i><a class="txt" href="#E1:29_1_2"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E2:29_1_2"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E2:29_1_2_1"/>
( <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> iff <font color="Maroon" title="c7">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</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="E4:29_1_2"/>
<font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X2</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</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>
