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<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">M0</font> be   <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">PTN</font>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="petri.html#NM4">transition</a> of <font color="Maroon">PTN</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">t</font> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 ;<br/><b>set </b><font color="Maroon">M</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <span class="p3">(<span class="default"><span class="p4"><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> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><i><font color="Green">E23</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">t</font></span><a href="finseq_1.html#K12">*&gt;</a></span> = 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <span class="p3">(<span class="default"><span class="p4"><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> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E3:19_1"/><i><font color="Green">E24</font></i>: 
<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> <a href="finseq_4.html#K4">/.</a> 1 <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><a NAME="E4:19_1"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <span class="p3">(<span class="default"><span class="p4"><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> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <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">t</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">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">t</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E2:19_1_2"/>
 <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">t</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><b>hence </b><a NAME="E3:19_1_2"/>
( <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> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <span class="p3">(<span class="default"><span class="p4"><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> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <span class="p3">(<span class="default"><span class="p4"><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> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">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">t</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="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="boolmark.html#K2" target="_self">Firing</a> <span class="p4">(<span class="default"><span class="p5"><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> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:19_1"/>
<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> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 <b>by </b><i>, <a class="ref" href="boolmark.html#T1" target="_self">Th1</a>, , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:19"/>
<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> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 ;<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="E4:19"/>
 <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">t</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="E5:19"/><i><font color="Green">E33</font></i>: 
<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> <a href="finseq_4.html#K4">/.</a> 1 <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">M0</font>
 <b>and </b><br/><a NAME="E6:19"/>
( <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">t</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">M0</font> &amp; ( 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">t</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/>( <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> <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="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> &amp; <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">t</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/>
<b>thus </b><a NAME="E7:19"/>
<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>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/>


</div>
