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

<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">T0</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c1">PTN</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K7" title="PETRI:func.7">*'</a> <font color="Olive" title="b1">T0</font> iff  ex <font color="Olive" title="b3">f</font> being  <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b4">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T0</font> &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/><b>let </b><font color="Maroon" title="c2">T0</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</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"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K7" title="PETRI:func.7">*'</a> <font color="Maroon" title="c2">T0</font> iff  ex <font color="Olive" title="b2">f</font> being  <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b3">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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#K7" title="PETRI:func.7">*'</a> <font color="Maroon" title="c2">T0</font> iff  ex <font color="Olive" title="b1">f</font> being  <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>





<b>thus </b><a NAME="E1:13"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K7" title="PETRI:func.7">*'</a> <font color="Maroon" title="c2">T0</font> implies  ex <font color="Olive" title="b1">f</font> being  <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">f</font> being  <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K7" title="PETRI:func.7">*'</a> <font color="Maroon" title="c2">T0</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K7" title="PETRI:func.7">*'</a> <font color="Maroon" title="c2">T0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being  <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">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="E3:13_1"/><i><font color="Green" title="E4">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">s</font>
 <b>and </b><br/><a NAME="E4:13_1"/><i><font color="Green" title="E5">A2</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</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">s</font>,<font color="Olive" title="b2">t</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#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font>, <font color="Maroon" title="c6">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="E6:13_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font>
 <b>and </b><br/><a NAME="E7:13_1"/><i><font color="Green" title="E7">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">s</font>,<font color="Maroon" title="c6">t</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#E4:13_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>take </b>
<font color="Maroon" title="c6">t</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<b>thus </b><a NAME="E8:13_1"/>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="petri.html#E3:13_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="petri.html#E6:13_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="petri.html#E7:13_1"><i><font color="Green" title="E7">A4</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>given </b><font color="Maroon" title="c4">f</font> being   <a href="petri.html#NM6" title="PETRI:NM.6">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font>, <font color="Maroon" title="c5">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><a NAME="E3:13"/><i><font color="Green" title="E4">A5</font></i>: 
( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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#K7" title="PETRI:func.7">*'</a> <font color="Maroon" title="c2">T0</font></span><br/>

<a NAME="E4:13"/>
<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#K1" title="PETRI:func.1">`1</a> 
 
<b>by </b><i><a class="txt" href="petri.html#E3:13"><i><font color="Green" title="E4">A5</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="E5:13"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K7" title="PETRI:func.7">*'</a> <font color="Maroon" title="c2">T0</font>
 <b>by </b><i><a class="txt" href="petri.html#E3:13"><i><font color="Green" title="E4">A5</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>
