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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font> implies ( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) holds <br/> <a href="prob_1.html#K2">Union</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) &amp; ( for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/>
<font color="Maroon">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:10_1"/>
for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) holds <br/> <a href="prob_1.html#K2">Union</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A1</font> as    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>
<a NAME="E3:10_1_1"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T46">PROB_1:46</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_1"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_1"/>
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) holds <br/> <a href="prob_1.html#K2">Union</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) &amp; ( for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 <b>by </b><i><a class="ref" href="prob_1.html#T32">PROB_1:32</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:10"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) holds <br/> <a href="prob_1.html#K2">Union</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) &amp; ( for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Olive">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 ;<br/>

<a NAME="E3:10"/>
for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:10_2"/><i><font color="Green">E48</font></i>: 
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) implies for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:10_2_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:10_2_1_1"/>
<font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:10_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:10_2_1_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D4">PROB_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_2_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:10_2"/>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) implies <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_2_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:10_2_2"/><b>then </b>
 <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:10_2_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_2"/>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#D5">PROB_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10"/>
<font color="Maroon">S</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T32">PROB_1:32</a></i>;<br/>


</div>
