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

<b>let </b><font color="Maroon">PTN</font> be    <a href="petri.html#L1" target="_self">PT_net_Str</a> ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="petri.html#NM4">transition</a> of <font color="Maroon">PTN</font>;<br/><b>let </b><font color="Maroon">S0</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="petri.html#U1" target="_self">Places</a> of <font color="Maroon">PTN</font>;<br/>





<b>thus </b><a NAME="E1:40"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font> <a href="petri.html#K6" target="_self">*'</a>  implies  <a href="petri.html#K7" target="_self">*'</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">S0</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font> <a href="petri.html#K6" target="_self">*'</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="petri.html#NM6">S-T_arc</a> of <font color="Maroon">PTN</font>, <font color="Maroon">s</font> being   <a href="petri.html#NM2">place</a> of <font color="Maroon">PTN</font><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:40_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:40_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K7" target="_self">*'</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:40_1"/>
<span class="p1">(<span class="default"><a href="petri.html#K7" target="_self">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


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

<b>assume </b><a NAME="E2:40"/>
<span class="p1">(<span class="default"><a href="petri.html#K7" target="_self">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="petri.html#NM2">place</a> of <font color="Maroon">PTN</font><b> such that </b><br/><a NAME="E4:40"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="petri.html#K7" target="_self">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S0</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:40"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K7" target="_self">*'</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="petri.html#NM6">S-T_arc</a> of <font color="Maroon">PTN</font>, <font color="Maroon">t0</font> being   <a href="petri.html#NM4">transition</a> of <font color="Maroon">PTN</font><b> such that </b><br/><a NAME="E7:40"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E8:40"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t0</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E9:40"/>
<font color="Maroon">t0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:40"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font> <a href="petri.html#K6" target="_self">*'</a> 
 <b>by </b><i>, </i>;<br/>


</div>
