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

<b>let </b><font color="Maroon" title="c1">N</font> be   <a href="net_1.html#NM1" title="NET_1:NM.1">Pnet</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
(  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="net_1.html#K2" title="NET_1:func.2">Pre</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="ref" href="net_1.html#T22" target="_self" title="NET_1:th.22">Th22</a></i>;<br/>
<a NAME="E3:24"/>
(  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green" title="E20">A3</font></i>: 
 <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:24_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="txt" href="net_1.html#E1:24"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E3:24_1"/><b>then </b>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:24_1"/>
 <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>
 <b>by </b><i><a class="txt" href="net_1.html#E1:24_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:24"/>
 <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>
 <b>by </b><i><a class="txt" href="net_1.html#E2:24"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="net_1.html#T16" target="_self" title="NET_1:th.16">Th16</a></i>;<br/>


</div>
