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





<b>thus </b><a NAME="E1:8"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K5" target="_self">*'</a> <font color="Maroon">S0</font> implies  ex <font color="Olive">f</font> being  <a href="petri.html#NM7">T-S_arc</a> of <font color="Maroon">PTN</font> ex <font color="Olive">s</font> being  <a href="petri.html#NM2">place</a> of <font color="Maroon">PTN</font> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">s</font></span><a href="tarski.html#K4">]</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K5" target="_self">*'</a> <font color="Maroon">S0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="petri.html#NM4">transition</a> of <font color="Maroon">PTN</font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E4:8_1"/><i><font color="Green">E19</font></i>: 
 ex <font color="Olive">f</font> being  <a href="petri.html#NM7">T-S_arc</a> of <font color="Maroon">PTN</font> ex <font color="Olive">s</font> being  <a href="petri.html#NM2">place</a> of <font color="Maroon">PTN</font> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">t</font>,<font color="Olive">s</font></span><a href="domain_1.html#K1">]</a></span> )
 ;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="petri.html#NM7">T-S_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="E6:8_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S0</font>
 <b>and </b><br/><a NAME="E7:8_1"/><i><font color="Green">E21</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">t</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E8:8_1"/>
( <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="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , </i>;<br/>


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

<b>given </b><font color="Maroon">f</font> being   <a href="petri.html#NM7">T-S_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><a NAME="E2:8"/><i><font color="Green">E23</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="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E3:8"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="petri.html#K3" target="_self">`1</a> 
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K5" target="_self">*'</a> <font color="Maroon">S0</font>
 <b>by </b><i/>;<br/>


</div>
