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

<b>let </b><font color="Maroon">N</font> be   <a href="net_1.html#NM1">Pnet</a>;<br/><b>let </b><font color="Maroon">x</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><b>that </b><br/><a NAME="E1:35"/>
 <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E3:35"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K12" target="_self">Output</a> <font color="Maroon">N</font>,<font color="Maroon">X</font> implies  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Olive">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K12" target="_self">Output</a> <font color="Maroon">N</font>,<font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="net_1.html#K10" target="_self">Ext</a> <font color="Maroon">N</font>,<font color="Maroon">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:35_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K10" target="_self">Ext</a> <font color="Maroon">N</font>,<font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:35_1"/>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Olive">z</font> )
 
<b>by </b><i><a class="ref" href="net_1.html#D8" target="_self">Def8</a>, </i>;<br/>
<b>hence </b><a NAME="E6:35_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Olive">y</font> )
 <b>by </b><i><a class="ref" href="net_1.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:35"/>
(  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Olive">y</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K12" target="_self">Output</a> <font color="Maroon">N</font>,<font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font><b> such that </b><a NAME="E1:35_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:35_2"/>
 <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K10" target="_self">Ext</a> <font color="Maroon">N</font>,<font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="net_1.html#D8" target="_self">Def8</a>, </i>;<br/>
<a NAME="E3:35_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="net_1.html#K10" target="_self">Ext</a> <font color="Maroon">N</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="net_1.html#D8" target="_self">Def8</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:35_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K12" target="_self">Output</a> <font color="Maroon">N</font>,<font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K12" target="_self">Output</a> <font color="Maroon">N</font>,<font color="Maroon">X</font> iff  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Olive">y</font> ) )
 <b>by </b><i/>;<br/>


</div>
