<?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="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E13</font></i>: 
 <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> 
 ;<br/>

<a NAME="E2:37"/><i><font color="Green">E17</font></i>: 
(  not <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> or ( <font color="Maroon">x</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> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> ) or  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="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Olive">y</font>,<font color="Maroon">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_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="E2:37_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/>
<a NAME="E3:37_1"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font> &amp; <font color="Olive">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/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:37_1"/><i><font color="Green">E26</font></i>: 
( <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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 ;<br/>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E7:37_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">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="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="net_1.html#D8" target="_self">Def8</a>, </i>;<br/>
<a NAME="E8:37_1"/><i><font color="Green">E29</font></i>: 
( ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> ) &amp; ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="net_1.html#K3" target="_self">Post</a> <font color="Maroon">N</font>,<font color="Maroon">z</font> ) )
 
<b>by </b><i><a class="txt" href="net_1.html#E7:37_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="net_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E9:37_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:37_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</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="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Olive">y</font>,<font color="Maroon">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:37_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U3" target="_self">Flow</a> of <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="net_1.html#D8" target="_self">Def8</a>, <a class="txt" href="net_1.html#E8:37_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="net_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E3:37_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Maroon">z</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="net_1.html#E7:37_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="net_1.html#T16" target="_self">Th16</a>, <a class="ref" href="net_1.html#T18" target="_self">Th18</a>, <a class="ref" href="net_1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:37_1"/>
( ( <font color="Maroon">x</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> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> ) or  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="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Olive">y</font>,<font color="Maroon">x</font> ) )
 <b>by </b><i><a class="ref" href="net_1.html#D8" target="_self">Def8</a>, <a class="txt" href="net_1.html#E7:37_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="net_1.html#E8:37_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="net_1.html#E9:37_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:37"/>
( ( ( <font color="Maroon">x</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> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> ) or  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="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Olive">y</font>,<font color="Maroon">x</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="37_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E48</font></i>: 
( ( <font color="Maroon">x</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> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> ) or  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="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Olive">y</font>,<font color="Maroon">x</font> ) )
 ;<br/>

<a NAME="E2:37_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</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> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</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="37_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</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> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E2:37_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="net_1.html#K5" target="_self">exit</a> <font color="Maroon">N</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="net_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:37_2_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E4:37_2_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <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>, , </i>;<br/>
<a NAME="E5:37_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:37_2_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>
 
<b>by </b><i><a class="ref" href="net_1.html#T16" target="_self">Th16</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:37_2_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/>


</div><b>end;</b></div>
<a NAME="E3:37_2"/>
(  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="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Olive">y</font>,<font color="Maroon">x</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="37_2_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="E1:37_2_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:37_2_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U3" target="_self">Flow</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="net_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:37_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K3" target="_self">Post</a> <font color="Maroon">N</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="net_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:37_2_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<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>
 
<b>by </b><i>, <a class="ref" href="net_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:37_2_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>, </i>;<br/>
<a NAME="E6:37_2_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="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:37_2_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="E4:37_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>
 <b>by </b><i><a class="ref" href="net_1.html#D8" target="_self">Def8</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:37"/>
( <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 ( ( <font color="Maroon">x</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> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">N</font> ) or  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="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">N</font> &amp;  <a href="net_1.html#R2" target="_self">post</a> <font color="Maroon">N</font>,<font color="Olive">y</font>,<font color="Maroon">x</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
