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<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">R3</font> be   <a href="pnproc_1.html#NM6">process</a> of <font color="Maroon">N</font>;<br/>









<b>thus </b><a NAME="E1:82"/>
<span class="p1">(<span class="default"><font color="Maroon">R1</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R2</font></span>)</span> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K12" target="_self">before</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</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="82_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="E1:82_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#K12" target="_self">before</a> <font color="Maroon">R2</font></span>)</span> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">C1</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Maroon">C2</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:82_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="lang1.html#K1">^</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R2</font> &amp; <font color="Maroon">C2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R3</font> )
 ;<br/>
<b>consider </b><font color="Maroon">fs1</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Maroon">fs2</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E5:82_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="lang1.html#K1">^</a> <font color="Maroon">fs2</font> &amp; <font color="Maroon">fs1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp; <font color="Maroon">fs2</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></i>;<br/>
<a NAME="E6:82_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="lang1.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">fs2</font> <a href="lang1.html#K1">^</a> <font color="Maroon">C2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E10"><i><font color="Green">Lemma50</font></i></a>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">C3</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E8:82_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">C3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs2</font> <a href="lang1.html#K1">^</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">fs2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> &amp; <font color="Maroon">C2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R3</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="E9:82_1"/>
<font color="Maroon">C3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="lang1.html#K1">^</a> <font color="Olive">a3</font></span>)</span> where <font color="Olive">a2</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Olive">a3</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> : ( <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> &amp; <font color="Olive">a3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R3</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E10:82_1"/><b>then </b>
<font color="Maroon">C3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font>
 
;<br/>
<a NAME="E11:82_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="lang1.html#K1">^</a> <font color="Olive">a1</font></span>)</span> where <font color="Olive">a</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Olive">a1</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp; <font color="Olive">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font> ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E12:82_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K12" target="_self">before</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font></span>)</span>
 ;<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="E2:82"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K12" target="_self">before</a> <span class="p1">(<span class="default"><font color="Maroon">R2</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">C1</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Maroon">C2</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:82"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="lang1.html#K1">^</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp; <font color="Maroon">C2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font> )
 ;<br/>
<b>consider </b><font color="Maroon">fs1</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Maroon">fs2</font> being   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:82"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="lang1.html#K1">^</a> <font color="Maroon">fs2</font> &amp; <font color="Maroon">fs1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> &amp; <font color="Maroon">fs2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R3</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="E7:82"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="lang1.html#K1">^</a> <font color="Maroon">fs1</font></span>)</span> <a href="lang1.html#K1">^</a> <font color="Maroon">fs2</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="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>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="E9:82"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="lang1.html#K1">^</a> <font color="Maroon">fs1</font> &amp; <font color="Maroon">C1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp; <font color="Maroon">fs1</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>, </i>;<br/>
<a NAME="E10:82"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="lang1.html#K1">^</a> <font color="Olive">b1</font></span>)</span> where <font color="Olive">a1</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Olive">b1</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> : ( <font color="Olive">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E11:82"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R2</font>
 
;<br/>
<a NAME="E12:82"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="lang1.html#K1">^</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">a</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font>, <font color="Olive">b</font> is   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">N</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R2</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R3</font> ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="pnproc_1.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E13:82"/>
<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#K12" target="_self">before</a> <font color="Maroon">R2</font></span>)</span> <a href="pnproc_1.html#K12" target="_self">before</a> <font color="Maroon">R3</font>
 ;<br/>


</div>
