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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="petri.html#L1">PT_net_Str</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:15"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="petri.html#V2">Deadlock-like</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>2</sub></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">b<sub>2</sub></font> being  <a href="petri.html#NM4">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>2</sub></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="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="petri.html#V2">Deadlock-like</a>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="petri.html#K5">*'</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:15_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>2</sub></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>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="petri.html#NM4">transition</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E4:15_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:15_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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="boolmark.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:15_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="margrel1.html#T38">MARGREL1:38</a>, <a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/>
<a NAME="E7:15_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></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="txt" href="boolmark.html#E5:15_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E8:15_1"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="petri.html#K7">*'</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="boolmark.html#E3:15_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="boolmark.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="petri.html#K5">*'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="boolmark.html#E2:15_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="petri.html#T19">PETRI:19</a></i>;<br/>
<a NAME="E10:15_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="petri.html#T20">PETRI:20</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <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">c<sub>4</sub></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">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="boolmark.html#E10:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="boolmark.html#T3" target="_self">Th3</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><a class="txt" href="boolmark.html#E3:15_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="boolmark.html#E8:15_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="boolmark.html#T3" target="_self">Th3</a></i>
;<br/>


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

<b>thus </b><a NAME="E2:15"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>2</sub></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">b<sub>2</sub></font> being  <a href="petri.html#NM4">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>2</sub></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> ) implies <font color="Maroon">c<sub>2</sub></font> is <a href="petri.html#V2">Deadlock-like</a> )
 <b>by </b><i><a class="txt" href="boolmark.html#E12"><i><font color="Green">Lemma9</font></i></a></i>;<br/>


</div>
