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<b>let </b><font color="Maroon" title="c1">PTN</font> be    <a href="petri.html#L1" title="PETRI:struct.1">PT_net_Str</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S0</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">PTN</font> holds   <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Olive" title="b1">S0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b2">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b2">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">S0</font> </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c2">S0</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">PTN</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span> </span><br/>



<b>thus </b><a NAME="E1:7"/>
 <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c4">t</font> being   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font><b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green" title="E1">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 <b>and </b><br/><a NAME="E4:7_1"/><i><font color="Green" title="E2">A2</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">s</font> being  <a href="petri.html#NM2" title="PETRI:NM.2">place</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">t</font>,<font color="Olive" title="b2">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font>, <font color="Maroon" title="c6">s</font> being   <a href="petri.html#NM2" title="PETRI:NM.2">place</a> of <font color="Maroon" title="c1">PTN</font><b> such that </b><br/><a NAME="E6:7_1"/><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font>
 <b>and </b><br/><a NAME="E7:7_1"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c6">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="petri.html#E4:7_1"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/>
<a NAME="E8:7_1"/>
( <font color="Maroon" title="c5">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> &amp; <font color="Maroon" title="c5">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> )
 
<b>by </b><i><a class="txt" href="petri.html#E7:7_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E9:7_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f0</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f0</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f0</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="petri.html#E3:7_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="petri.html#E6:7_1"><i><font color="Green" title="E3">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span>  or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font> )</span><br/>

<b>assume </b><a NAME="E2:7"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span> where <font color="Olive" title="b1">f</font> is   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font> : <font color="Olive" title="b1">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">T-S_arc</a> of <font color="Maroon" title="c1">PTN</font><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green" title="E1">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> 
 <b>and </b><br/><a NAME="E5:7"/><i><font color="Green" title="E2">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S0</font>
 ;<br/>
<a NAME="E6:7"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="petri.html#K3" title="PETRI:func.3">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="petri.html#K4" title="PETRI:func.4">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>hence </b><a NAME="E7:7"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">S0</font>
 <b>by </b><i><a class="txt" href="petri.html#E4:7"><i><font color="Green" title="E1">A5</font></i></a>, <a class="txt" href="petri.html#E5:7"><i><font color="Green" title="E2">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
