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

<b>let </b><font color="Maroon">P</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">N</font> be    <a href="pnproc_1.html#M3" target="_self">Petri_net</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">R1</font> be   <a href="pnproc_1.html#NM6">process</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">R2</font> be   <a href="pnproc_1.html#NM6">process</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="pnproc_1.html#NM6">process</a> of <font color="Maroon">N</font>;<br/>









<b>thus </b><a NAME="E1:79"/>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:79_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> :  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:79_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">q1</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a>, <font color="Maroon">q2</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E6:79_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E7:79_1"/>
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> or  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:79_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C1</font> where <font color="Olive">C1</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> :  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C2</font> where <font color="Olive">C2</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> :  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T7" target="_self">Th7</a>, <a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>;<br/>
<a NAME="E9:79_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font> )
 
;<br/>
<b>hence </b><a NAME="E10:79_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:79"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font> )
 </span><b>by </b><i><a class="ref" href="pnproc_1.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1"><b>suppose </b></a><a NAME="E1:79_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:79_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> :  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:79_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 ;<br/><b>consider </b><font color="Maroon">q1</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a>, <font color="Maroon">q2</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E6:79_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>;<br/><a NAME="E7:79_2_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:79_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C1</font> where <font color="Olive">C1</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> :  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a>, </i>;<br/><b>hence </b><a NAME="E9:79_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_2"><b>suppose </b></a><a NAME="E1:79_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:79_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C</font> where <font color="Olive">C</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> :  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:79_2_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 ;<br/><b>consider </b><font color="Maroon">q1</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a>, <font color="Maroon">q2</font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E6:79_2_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a></i>;<br/><a NAME="E7:79_2_2"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:79_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">C2</font> where <font color="Olive">C2</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> :  ex <font color="Olive">q1</font>, <font color="Olive">q2</font> being  <a href="finseq_1.html#NM4">FinSubsequence</a> st <br/>( <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <font color="Olive">q1</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">q2</font> &amp; <font color="Olive">q1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">q2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a>, </i>;<br/><b>hence </b><a NAME="E9:79_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">R2</font></span>)</span> <a href="pnproc_1.html#K13" target="_self">concur</a> <font color="Maroon">R</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
