<?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">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E33</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> ) )
 ;<br/>

<a NAME="E2:28"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">A1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:28_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">A1</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:28_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1" target="_self">.</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , </i>;<br/>


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

<b>assume </b><a NAME="E2:28_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E3:28_1"/><b>then </b>
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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:28"/>
 <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="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
