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

<b>consider </b><font color="Maroon" title="c4">O12</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>, <font color="Maroon" title="c5">O21</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font><b> such that </b><br/><a NAME="E2:28_1_1"/><i><font color="Green" title="E18">P1</font></i>: 
<font color="Maroon" title="c3">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c5">O21</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="petri_2.html#D12" target="_self" title="PETRI_2:def.12">Def8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">TO1</font> =  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">TO2</font> =  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font>;<br/>
<b>set </b><font color="Maroon" title="c8">K1</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">K2</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">L1</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">L2</font> =  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">Y1</font> =  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">Y2</font> =  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">t01</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">CPNT1</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t01</font> &amp; $2 is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="Olive" title="b1">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Olive" title="b1">t01</font></span>)</span> );<br/>
<a NAME="E3:28_1_1"/><i><font color="Green" title="E19">P2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c1">CPNT1</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">x</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="c1">CPNT1</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_1"/>
<font color="Maroon" title="c14">x</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="c1">CPNT1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c15">t01</font> being   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">CPNT1</font><b> such that </b><br/><a NAME="E3:28_1_1_1"/><i><font color="Green" title="E20">P21</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t01</font> &amp; <font color="Maroon" title="c15">t01</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> )
 ;<br/>
<b>set </b><font color="Maroon" title="c16">t1</font> =  <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="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c17">t2</font> =  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font>;<br/>
<b>consider </b><font color="Maroon" title="c18">y</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon" title="c18">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c18">y</font>] )</span><br/>

<b>set </b><font color="Maroon" title="c19">H1</font> =  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c20">H2</font> =  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span>;<br/>
<a NAME="E5:28_1_1_1"/><i><font color="Green" title="E21">XH1</font></i>: 
 <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="petri_2.html#E6"><i><font color="Green" title="E4">LmThin</font></i></a></i>;<br/>
<a NAME="E6:28_1_1_1"/><i><font color="Green" title="E22">XH2</font></i>: 
 <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="petri_2.html#E6"><i><font color="Green" title="E4">LmThin</font></i></a></i>;<br/>
<a NAME="E7:28_1_1_1"/><i><font color="Green" title="E23">J1</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E8:28_1_1_1"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T141" title="FUNCT_2:th.141">FUNCT_2:141</a></i>;<br/>
<a NAME="E9:28_1_1_1"/><b>then </b><i><font color="Green" title="E24">J2</font></i>: 
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:28_1_1_1"><i><font color="Green" title="E23">J1</font></i></a></i>;<br/>
<a NAME="E10:28_1_1_1"/>
 <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">t01</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#T128" title="PARTFUN1:th.128">PARTFUN1:128</a>, <a class="txt" href="#E5:28_1_1_1"><i><font color="Green" title="E21">XH1</font></i></a>, <a class="txt" href="#E6:28_1_1_1"><i><font color="Green" title="E22">XH2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:28_1_1_1"/>
( <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c18">y</font>] )
 <b>by </b><i><a class="txt" href="#E3:28_1_1_1"><i><font color="Green" title="E20">P21</font></i></a>, <a class="txt" href="#E9:28_1_1_1"><i><font color="Green" title="E24">J2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">q12</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font>, <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:28_1_1"/><i><font color="Green" title="E20">P3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c1">CPNT1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c14">q12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:28_1_1"><i><font color="Green" title="E19">P2</font></i></a>);<br/></i>
<div><i><font color="Green" title="E21">P4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">tt01</font> be   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">CPNT1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">tt01</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> implies <font color="Maroon" title="c14">q12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">tt01</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="c15">tt01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">tt01</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:28_1_1_2"/>
<font color="Maroon" title="c15">tt01</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">q12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">tt01</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="c15">tt01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">tt01</font></span>)</span></span><br/><a NAME="E2:28_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">tt01</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="c1">CPNT1</font>
 ;<br/><a NAME="E3:28_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">t01</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">CPNT1</font> st <br/>( <font color="Maroon" title="c15">tt01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t01</font> &amp; <font color="Maroon" title="c14">q12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">tt01</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="Olive" title="b1">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Olive" title="b1">t01</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E5:28_1_1"><i><font color="Green" title="E20">P3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_2"/>
<font color="Maroon" title="c14">q12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">tt01</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="c15">tt01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c4">O12</font>,<font color="Maroon" title="c15">tt01</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">t02</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c2">CPNT2</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t02</font> &amp; $2 is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="Olive" title="b1">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Olive" title="b1">t02</font></span>)</span> );<br/>
<a NAME="E7:28_1_1"/><i><font color="Green" title="E22">R2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c2">CPNT2</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">x</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="c2">CPNT2</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_3"/>
<font color="Maroon" title="c15">x</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="c2">CPNT2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c16">t02</font> being   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c2">CPNT2</font><b> such that </b><br/><a NAME="E3:28_1_1_3"/><i><font color="Green" title="E23">R21</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">t02</font> &amp; <font color="Maroon" title="c16">t02</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> )
 ;<br/>
<b>set </b><font color="Maroon" title="c17">t1</font> =  <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="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c18">t2</font> =  <a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font>;<br/>
<b>consider </b><font color="Maroon" title="c19">y</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon" title="c19">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c19">y</font>] )</span><br/>

<b>set </b><font color="Maroon" title="c20">H1</font> =  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c21">H2</font> =  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span>;<br/>
<a NAME="E5:28_1_1_3"/><i><font color="Green" title="E24">XH1</font></i>: 
 <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="petri_2.html#E6"><i><font color="Green" title="E4">LmThin</font></i></a></i>;<br/>
<a NAME="E6:28_1_1_3"/><i><font color="Green" title="E25">XH2</font></i>: 
 <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="petri_2.html#E6"><i><font color="Green" title="E4">LmThin</font></i></a></i>;<br/>
<a NAME="E7:28_1_1_3"/><i><font color="Green" title="E26">J1</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E8:28_1_1_3"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T141" title="FUNCT_2:th.141">FUNCT_2:141</a></i>;<br/>
<a NAME="E9:28_1_1_3"/><b>then </b><i><font color="Green" title="E27">J2</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:28_1_1_3"><i><font color="Green" title="E26">J1</font></i></a></i>;<br/>
<a NAME="E10:28_1_1_3"/>
 <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p2">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">t02</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#T128" title="PARTFUN1:th.128">PARTFUN1:128</a>, <a class="txt" href="#E5:28_1_1_3"><i><font color="Green" title="E24">XH1</font></i></a>, <a class="txt" href="#E6:28_1_1_3"><i><font color="Green" title="E25">XH2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:28_1_1_3"/>
( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c19">y</font>] )
 <b>by </b><i><a class="txt" href="#E3:28_1_1_3"><i><font color="Green" title="E23">R21</font></i></a>, <a class="txt" href="#E9:28_1_1_3"><i><font color="Green" title="E27">J2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">q21</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font>, <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E9:28_1_1"/><i><font color="Green" title="E23">R3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c2">CPNT2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c15">q21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E7:28_1_1"><i><font color="Green" title="E22">R2</font></i></a>);<br/></i>
<div><i><font color="Green" title="E24">R4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">tt02</font> be   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c2">CPNT2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">tt02</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> implies <font color="Maroon" title="c15">q21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">tt02</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="c16">tt02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">tt02</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:28_1_1_4"/>
<font color="Maroon" title="c16">tt02</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">q21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">tt02</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="c16">tt02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">tt02</font></span>)</span></span><br/><a NAME="E2:28_1_1_4"/><b>then </b>
<font color="Maroon" title="c16">tt02</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="c2">CPNT2</font>
 ;<br/><a NAME="E3:28_1_1_4"/><b>then </b>
 ex <font color="Olive" title="b1">t02</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c2">CPNT2</font> st <br/>( <font color="Maroon" title="c16">tt02</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t02</font> &amp; <font color="Maroon" title="c15">q21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">tt02</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="Olive" title="b1">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Olive" title="b1">t02</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E9:28_1_1"><i><font color="Green" title="E23">R3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_4"/>
<font color="Maroon" title="c15">q21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">tt02</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="c16">tt02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Maroon" title="c5">O21</font>,<font color="Maroon" title="c16">tt02</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>take </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">q12</font>,<font color="Maroon" title="c15">q21</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">q12</font>, <font color="Olive" title="b2">q21</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ex <font color="Olive" title="b3">O12</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font> ex <font color="Olive" title="b4">O21</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font> st <br/>( <font color="Maroon" title="c3">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">O12</font>,<font color="Olive" title="b4">O21</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">q12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font> &amp; ( for <font color="Olive" title="b5">t01</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">CPNT1</font>  st <font color="Olive" title="b5">t01</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> holds <br/><font color="Olive" title="b1">q12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">t01</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="Olive" title="b5">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Olive" title="b3">O12</font>,<font color="Olive" title="b5">t01</font></span>)</span> ) &amp; ( for <font color="Olive" title="b5">t02</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c2">CPNT2</font>  st <font color="Olive" title="b5">t02</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> holds <br/><font color="Olive" title="b2">q21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">t02</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="Olive" title="b5">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Olive" title="b4">O21</font>,<font color="Olive" title="b5">t02</font></span>)</span> ) &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">q12</font>,<font color="Maroon" title="c15">q21</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">q12</font>,<font color="Olive" title="b2">q21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E11:28_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">q12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:28_1_1"/>
 ex <font color="Olive" title="b1">q12</font>, <font color="Olive" title="b2">q21</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ex <font color="Olive" title="b3">O12</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font>,the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c2">CPNT2</font> ex <font color="Olive" title="b4">O21</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font>,the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">CPNT1</font> st <br/>( <font color="Maroon" title="c3">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">O12</font>,<font color="Olive" title="b4">O21</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">q12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c1">CPNT1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="petri_2.html#K7" title="PETRI_2:func.7">Outbds</a> <font color="Maroon" title="c2">CPNT2</font> &amp; ( for <font color="Olive" title="b5">t01</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">CPNT1</font>  st <font color="Olive" title="b5">t01</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> holds <br/><font color="Olive" title="b1">q12</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">t01</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<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="Olive" title="b5">t01</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c1">CPNT1</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Olive" title="b3">O12</font>,<font color="Olive" title="b5">t01</font></span>)</span> ) &amp; ( for <font color="Olive" title="b5">t02</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c2">CPNT2</font>  st <font color="Olive" title="b5">t02</font> is <a href="petri_2.html#V2" title="PETRI_2:attr.2">outbound</a> holds <br/><font color="Olive" title="b2">q21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">t02</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<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="Olive" title="b5">t02</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>, <a href="petri_2.html#K2" title="PETRI_2:func.2">thin_cylinders</a> the <a href="petri_2.html#U1" title="PETRI_2:sel.1">ColoredSet</a> of <font color="Maroon" title="c2">CPNT2</font>,<span class="p1">(<span class="default"><a href="relset_2.html#K1" title="RELSET_2:func.1">Im</a> <font color="Olive" title="b4">O21</font>,<font color="Olive" title="b5">t02</font></span>)</span> ) &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">q12</font>,<font color="Maroon" title="c15">q21</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">q12</font>,<font color="Olive" title="b2">q21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E2:28_1_1"><i><font color="Green" title="E18">P1</font></i></a>, <a class="txt" href="#E6:28_1_1"><i><font color="Green" title="E21">P4</font></i></a>, <a class="txt" href="#E10:28_1_1"><i><font color="Green" title="E24">R4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
