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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="net_1.html#NM1">Pnet</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U2" target="_self">Transitions</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K2" target="_self">Pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> implies  <a href="net_1.html#R1" target="_self">pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K2" target="_self">Pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:20_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U3" target="_self">Flow</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="net_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
 <a href="net_1.html#R1" target="_self">pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="net_1.html#E1:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="net_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20"/>
(  <a href="net_1.html#R1" target="_self">pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K2" target="_self">Pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/>
 <a href="net_1.html#R1" target="_self">pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:20_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U3" target="_self">Flow</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="net_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:20_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="net_1.html#U1" target="_self">Places</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="net_1.html#E1:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="net_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E4:20_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K1" target="_self">Elements</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="net_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:20_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K2" target="_self">Pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="net_1.html#E2:20_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="net_1.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="net_1.html#K2" target="_self">Pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> iff  <a href="net_1.html#R1" target="_self">pre</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="net_1.html#E2:20"><i><font color="Green">E16</font></i></a></i>;<br/>


</div>
