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

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Y</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E3:33"/>
<font color="Maroon">Y</font> is <a href="finsub_1.html#V2">cap-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a><br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">B'</font> = <font color="Maroon">B</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A1</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:33_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">A1</font> <a href="prob_1.html#K1" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Maroon">A1</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:33_1"/><i><font color="Green">E36</font></i>: 
 <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:33_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  <font color="Maroon">A1</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:33_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
<font color="Maroon">Y</font> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div>
