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

<span class="kw">set </span><font color="Maroon" title="c1">PLA</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">red</font>, <font color="Maroon" title="c5">yellow</font>, <font color="Maroon" title="c6">blue</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">TRA</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">TSA</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E2:24_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">TSA</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">STA</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E4:24_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">STA</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">CS</font> = <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c4">red</font>,<font color="Maroon" title="c5">yellow</font>,<font color="Maroon" title="c6">blue</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">fa</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <span class="p2"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c4">red</font>,<font color="Maroon" title="c5">yellow</font>,<font color="Maroon" title="c6">blue</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <span class="p2"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c4">red</font>,<font color="Maroon" title="c5">yellow</font>,<font color="Maroon" title="c6">blue</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">f</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">fa</font>;<br/>
<span class="kw">take </span><font color="Maroon" title="c15">CPNT</font> =  <a href="petri_2.html#G1" title="PETRI_2:aggr.1">Colored_PT_net_Str</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c11">STA</font>,<font color="Maroon" title="c9">TSA</font>,<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c4">red</font>,<font color="Maroon" title="c5">yellow</font>,<font color="Maroon" title="c6">blue</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">fa</font></span>)</span> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">CPNT</font> is  <a href="petri_2.html#V1" title="PETRI_2:attr.1">strict</a>  &amp; <font color="Maroon" title="c15">CPNT</font> is  <a href="petri_2.html#V3" title="PETRI_2:attr.3">Colored-PT-net-like</a>  )</span><br/>

<div><span class="lab"><font color="Green" title="E16">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:24_1_1_1"/>
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c4">red</font>,<font color="Maroon" title="c5">yellow</font>,<font color="Maroon" title="c6">blue</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c4">red</font>,<font color="Maroon" title="c5">yellow</font>,<font color="Maroon" title="c6">blue</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">CS1</font> = <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c4">red</font>,<font color="Maroon" title="c5">yellow</font>,<font color="Maroon" title="c6">blue</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c15">CPNT</font> ;<br/><span class="kw">let </span><font color="Maroon" title="c17">t</font> be   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c15">CPNT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font> implies  ex <font color="Olive" title="b1">CS1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c15">CPNT</font> ex <font color="Olive" title="b2">I</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ex <font color="Olive" title="b3">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="petri.html#K8" title="PETRI:func.8">*'</a> </span>)</span> st the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Olive" title="b1">CS1</font>,<font color="Olive" title="b2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Olive" title="b1">CS1</font>,<font color="Olive" title="b3">O</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c17">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">CS1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c15">CPNT</font> ex <font color="Olive" title="b2">I</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ex <font color="Olive" title="b3">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="petri.html#K8" title="PETRI:func.8">*'</a> </span>)</span> st the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Olive" title="b1">CS1</font>,<font color="Olive" title="b2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Olive" title="b1">CS1</font>,<font color="Olive" title="b3">O</font></span>)</span></span><br/><a NAME="E4:24_1_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:24_1_1_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:24_1_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E7:24_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">TSA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E8:24_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="petri.html#K8" title="PETRI:func.8">*'</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="petri.html#T8" title="PETRI:th.8">PETRI:8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">O</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="petri.html#K8" title="PETRI:func.8">*'</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E10:24_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">STA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E5:24_1_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:24_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="petri.html#T6" title="PETRI:th.6">PETRI:6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">I</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E13:24_1_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c13">fa</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c16">CS1</font>,<font color="Maroon" title="c19">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Maroon" title="c16">CS1</font>,<font color="Maroon" title="c18">O</font></span>)</span>
 ;<br/><a NAME="E14:24_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">fa</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E15:24_1_1_1"/>
 ex <font color="Olive" title="b1">CS1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c15">CPNT</font> ex <font color="Olive" title="b2">I</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ex <font color="Olive" title="b3">O</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="petri.html#K8" title="PETRI:func.8">*'</a> </span>)</span> st the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">t</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Olive" title="b1">CS1</font>,<font color="Olive" title="b2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> <font color="Olive" title="b1">CS1</font>,<font color="Olive" title="b3">O</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24_1_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:24_1_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c13">fa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c16">a1</font> = <font color="Maroon" title="c2">a</font> as   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c15">CPNT</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font> implies <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c15">CPNT</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c15">CPNT</font></span>)</span> )</span><br/><a NAME="E2:24_1_1_2"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:24_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">a1</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">TSA</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E4:24_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
not <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">a1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="petri.html#K8" title="PETRI:func.8">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="petri.html#T8" title="PETRI:th.8">PETRI:8</a></span>;<br/><a NAME="E5:24_1_1_2"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
not <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c15">CPNT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1_1_2_1"/>
<font color="Maroon" title="c16">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c15">CPNT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:24_1_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c15">CPNT</font> st <br/>( <font color="Maroon" title="c16">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> is  <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_1_2"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="petri_2.html#D8" target="_self" title="PETRI_2:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E6:24_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c15">CPNT</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c15">CPNT</font></span>)</span></span><br/><a NAME="E7:24_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:24_1_1_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c15">CPNT</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c15">CPNT</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_1_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:24_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="petri_2.html#U2" title="PETRI_2:sel.2">firing-rule</a> of <font color="Maroon" title="c15">CPNT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c15">CPNT</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c15">CPNT</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:24_1_1"/>
( <font color="Maroon" title="c15">CPNT</font> is  <a href="petri_2.html#V1" title="PETRI_2:attr.1">strict</a>  &amp; <font color="Maroon" title="c15">CPNT</font> is  <a href="petri_2.html#V3" title="PETRI_2:attr.3">Colored-PT-net-like</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="petri_2.html#D10" target="_self" title="PETRI_2:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
