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

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">X2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff ( <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ) )
 <b>and </b><br/><a NAME="E2:31_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff ( <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ) )
 ;<br/>

<a NAME="E3:31_1_2"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:31_1_2_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> implies <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:31_1_2_1_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:31_1_2_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:31_1_2_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> implies <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1_2_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E2:31_1_2_1_2"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp; <font color="Maroon">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:31_1_2_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_1_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
