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

<b>let </b><font color="Maroon">PTN</font> be    <a href="petri.html#L1">PT_net_Str</a> ;<br/><b>let </b><font color="Maroon">Sd</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font>;<br/>



<b>thus </b><a NAME="E1:21"/>
( <font color="Maroon">Sd</font> is <a href="petri.html#V2">Deadlock-like</a> implies for <font color="Olive">M0</font> being  <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">PTN</font>  st <font color="Olive">M0</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span> holds <br/>for <font color="Olive">Q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">PTN</font>  st <font color="Olive">Q</font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Olive">M0</font> holds <br/><span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Olive">Q</font>,<font color="Olive">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">Sd</font> is <a href="petri.html#V2">Deadlock-like</a>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="petri.html#K5">*'</a> <font color="Maroon">Sd</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="petri.html#K6">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="petri.html#D5">PETRI:def 5</a></i>;<br/>
<b>let </b><font color="Maroon">M0</font> be   <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">PTN</font>;<br/>

<b>assume </b><a NAME="E3:21_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">M0</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">PTN</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">M0</font> implies <span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span> );<br/>
<a NAME="E4:21_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">PTN</font>]
 
<b>by </b><i><a class="ref" href="boolmark.html#T1" target="_self">Th1</a>, </i>;<br/>
<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">PTN</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="petri.html#NM4">transition</a> of <font color="Maroon">PTN</font>;<br/><b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Q</font>]
 ;<br/><b>thus </b><a NAME="E2:21_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Q</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 ;<br/>

<a NAME="E2:21_1_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="boolmark.html#R2" target="_self">is_firable_on</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:21_1_1_1"/>
 <a href="boolmark.html#K3" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">M0</font> <a href="funct_2.html#R2">=</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">M</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="boolmark.html#K1" target="_self">Bool_marks_of</a> <font color="Maroon">PTN</font><b> such that </b><br/><a NAME="E5:21_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E6:21_1_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="boolmark.html#K3" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">M0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span>
 <b>and </b><br/><a NAME="E7:21_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font></span>)</span>
 <b>and </b><br/><a NAME="E8:21_1_1_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E9:21_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>
<a NAME="E10:21_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="boolmark.html#K3" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">M0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="petri.html#K8">*'</a> </span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="boolmark.html#E2:21_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="boolmark.html#E3:21_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="boolmark.html#E4:21_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E11:21_1_1_1"/>
<font color="Maroon">x</font> <a href="boolmark.html#R1" target="_self">is_firable_on</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:21_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:21_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/>
<a NAME="E14:21_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="boolmark.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E15:21_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="petri.html#K7">*'</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Sd</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E16:21_1_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K5">*'</a> <font color="Maroon">Sd</font>
 
<b>by </b><i>, <a class="ref" href="petri.html#T19">PETRI:19</a></i>;<br/>
<a NAME="E17:21_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="petri.html#K8">*'</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Sd</font>
 
<b>by </b><i><a class="ref" href="petri.html#T20">PETRI:20</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font>
<b>by </b><i><a class="txt" href="boolmark.html#E5:21_1"><i><font color="Green">E25</font></i></a>, </i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, , , , </i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E6:21_1"/>
for <font color="Olive">Q0</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">PTN</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Q0</font>]
 <b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );</i><br/>


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

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">M0</font> being  <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">PTN</font>  st <font color="Olive">M0</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span> holds <br/>for <font color="Olive">Q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">PTN</font>  st <font color="Olive">Q</font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Olive">M0</font> holds <br/><span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Olive">Q</font>,<font color="Olive">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>assume </b><a NAME="E3:21"/>
 <a href="petri.html#K5">*'</a> <font color="Maroon">Sd</font> is  not   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="petri.html#K6">*'</a> </span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="petri.html#D5">PETRI:def 5</a><br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="petri.html#NM4">transition</a> of <font color="Maroon">PTN</font><b> such that </b><br/><a NAME="E5:21"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K5">*'</a> <font color="Maroon">Sd</font>
 <b>and </b><br/><a NAME="E6:21"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sd</font> <a href="petri.html#K6">*'</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">M0</font> = <span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span>;<br/>
<a NAME="E7:21"/><i><font color="Green">E45</font></i>: 
 <a href="subset_1.html#K2">[#]</a> the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="hidden.html#R1">=</a> the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E8:21"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Sd</font> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then </b><i><font color="Green">E46</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> = 
the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Sd</font>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font>
<b>by </b><i>, <a class="ref" href="subset_1.html#T28">SUBSET_1:28</a></i>
;<br/>
<a NAME="E10:21"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E11:21"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:21"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E13:21"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E14:21"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:21"/><b>then </b>
<span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font>,<a href="margrel1.html#K6">BOOLEAN</a> 
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M0</font> = <span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">Sd</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> as   <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">PTN</font> ;<br/>
<a NAME="E17:21"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">M0</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">t</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">PTN</font> ;<br/>
<a NAME="E19:21"/>
<font color="Maroon">Sd</font> <a href="xboole_0.html#R1">misses</a>  <a href="petri.html#K7">*'</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="petri.html#T19">PETRI:19</a></i>;<br/>
<a NAME="E20:21"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M0</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E21:21"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E22:21"/>
<span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="petri.html#U1">Places</a> of <font color="Maroon">PTN</font> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T144">RELAT_1:144</a></i>;<br/>
<a NAME="E23:21"/><b>then </b>
<font color="Maroon">M0</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7">*'</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E24:21"/><b>then </b>
<font color="Maroon">t</font> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E25:21"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">Q</font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E26:21"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="petri.html#K8">*'</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Sd</font>
 
<b>by </b><i><a class="ref" href="boolmark.html#T1" target="_self">Th1</a>, <a class="ref" href="petri.html#T20">PETRI:20</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E28:21"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="petri.html#K8">*'</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Sd</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E29:21"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> <a href="petri.html#K8">*'</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sd</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <font color="Maroon">t</font>,<font color="Maroon">M0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E31:21"/><b>then </b>
 <a href="margrel1.html#K8">TRUE</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <font color="Maroon">t</font>,<font color="Maroon">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font>
 
<b>by </b><i><a class="ref" href="boolmark.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E32:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <font color="Maroon">t</font>,<font color="Maroon">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E33:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">Q</font>,<font color="Maroon">M0</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Sd</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E34:21"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div>
