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<b>let </b><font color="Maroon" title="c1">P</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"> for <font color="Olive" title="b1">N</font> being   <a href="pnproc_1.html#M4" title="PNPROC_1:mode.4">Petri_net</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b2">R1</font>, <font color="Olive" title="b3">R2</font>, <font color="Olive" title="b4">R3</font> being  <a href="pnproc_1.html#NM7" title="PNPROC_1:NM.7">process</a> of <font color="Olive" title="b1">N</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Olive" title="b3">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Olive" title="b4">R3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Olive" title="b4">R3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">N</font> be    <a href="pnproc_1.html#M4" title="PNPROC_1:mode.4">Petri_net</a> of <font color="Maroon" title="c1">P</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">R1</font>, <font color="Olive" title="b2">R2</font>, <font color="Olive" title="b3">R3</font> being  <a href="pnproc_1.html#NM7" title="PNPROC_1:NM.7">process</a> of <font color="Maroon" title="c2">N</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Olive" title="b2">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Olive" title="b3">R3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Olive" title="b3">R3</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">R2</font>, <font color="Maroon" title="c5">R3</font> be   <a href="pnproc_1.html#NM7" title="PNPROC_1:NM.7">process</a> of <font color="Maroon" title="c2">N</font>; <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"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span>)</span></span><br/>









<b>thus </b><a NAME="E1:82"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</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"> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span><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" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:82_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c7">C1</font>, <font color="Maroon" title="c8">C2</font> being   <a href="pnproc_1.html#NM6" title="PNPROC_1:NM.6">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:82_1"/><i><font color="Green" title="E50">A1</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c8">C2</font> &amp; <font color="Maroon" title="c7">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c8">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R3</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">fs1</font>, <font color="Maroon" title="c10">fs2</font> being   <a href="pnproc_1.html#NM6" title="PNPROC_1:NM.6">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E5:82_1"/><i><font color="Green" title="E51">A2</font></i>: 
( <font color="Maroon" title="c7">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c10">fs2</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Maroon" title="c10">fs2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:82_1"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<a NAME="E6:82_1"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">fs2</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c8">C2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E3:82_1"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="pnproc_1.html#E5:82_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">C3</font> being   <a href="pnproc_1.html#NM6" title="PNPROC_1:NM.6">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E8:82_1"/><i><font color="Green" title="E53">A4</font></i>: 
( <font color="Maroon" title="c11">C3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">fs2</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c8">C2</font> &amp; <font color="Maroon" title="c10">fs2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c8">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R3</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:82_1"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="pnproc_1.html#E5:82_1"><i><font color="Green" title="E51">A2</font></i></a></i>;<br/>
<a NAME="E9:82_1"/>
<font color="Maroon" title="c11">C3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E8:82_1"><i><font color="Green" title="E53">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:82_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span>)</span>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E5:82_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="pnproc_1.html#E6:82_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="pnproc_1.html#E8:82_1"><i><font color="Green" title="E53">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>let </b><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span>)</span> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font> )</span><br/>

<b>assume </b><a NAME="E2:82"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">C1</font>, <font color="Maroon" title="c8">C2</font> being   <a href="pnproc_1.html#NM6" title="PNPROC_1:NM.6">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E4:82"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c8">C2</font> &amp; <font color="Maroon" title="c7">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Maroon" title="c8">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">fs1</font>, <font color="Maroon" title="c10">fs2</font> being   <a href="pnproc_1.html#NM6" title="PNPROC_1:NM.6">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E6:82"/><i><font color="Green" title="E51">A6</font></i>: 
( <font color="Maroon" title="c8">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c10">fs2</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c10">fs2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">R3</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E4:82"><i><font color="Green" title="E50">A5</font></i></a></i>;<br/>
<a NAME="E7:82"/><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">fs1</font></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c10">fs2</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E4:82"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E6:82"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">C</font> being   <a href="pnproc_1.html#NM6" title="PNPROC_1:NM.6">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E9:82"/><i><font color="Green" title="E53">A8</font></i>: 
( <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">fs1</font> &amp; <font color="Maroon" title="c7">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E4:82"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E6:82"><i><font color="Green" title="E51">A6</font></i></a></i>;<br/>
<a NAME="E10:82"/>
<font color="Maroon" title="c11">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E9:82"><i><font color="Green" title="E53">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:82"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K10" title="PNPROC_1:func.10">before</a> <font color="Maroon" title="c5">R3</font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E6:82"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="pnproc_1.html#E7:82"><i><font color="Green" title="E52">A7</font></i></a>, <a class="txt" href="pnproc_1.html#E9:82"><i><font color="Green" title="E53">A8</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/>


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