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<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>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>assume </b><a NAME="E1:78_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</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">R2</font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E2:78_1_2"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</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">R1</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="78_1_2_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="E1:78_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</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><b> such that </b><br/><a NAME="E3:78_1_2_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</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">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">R2</font> ) )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E8"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <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">R1</font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E4:78_1_2_1"/>
<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">R1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="pnproc_1.html#T7" target="_self">Th7</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="E3:78_1_2"/>
<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">R1</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><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="E5:78_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</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">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">R1</font> ) )
 ;<br/>
<b>thus </b><a NAME="E6:78_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E8"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="pnproc_1.html#T7" target="_self">Th7</a></i>;<br/>


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