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

<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <a href="net_1.html#NM1" title="NET_1:NM.1">Pnet</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font></span>)</span><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>  st  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">X</font> iff ( ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>  st  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> iff ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> iff ( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) ) ) )</span><br/>





<a NAME="E1:43"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
( ( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) ) implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
(  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c4">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><a NAME="E2:43_1_1"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E3:43_1_1"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font>
 <span class="kw">and </span><br/><a NAME="E4:43_1_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E5:43_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U3" title="NET_1:sel.3">Flow</a> of <font color="Maroon" title="c1">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="net_1.html#D6" target="_self" title="NET_1:def.6">Def6</a></span>;<br/>
<a NAME="E6:43_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K3" title="NET_1:func.3">Post</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="net_1.html#D8" target="_self" title="NET_1:def.8">Def8</a></span>;<br/>
<a NAME="E7:43_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K5" title="NET_1:func.5">exit</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="net_1.html#D10" target="_self" title="NET_1:def.10">Def10</a></span>;<br/>
<a NAME="E8:43_1_1"/>
 <a href="net_1.html#K5" title="NET_1:func.5">exit</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K10" title="NET_1:func.10">Ext</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="net_1.html#T29" target="_self" title="NET_1:th.29">Th29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:43_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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="E2:43_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_1_2"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E2:43_1_2"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E3:43_1_2"/>
(  <a href="net_1.html#K5" title="NET_1:func.5">exit</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">x</font> <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="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="net_1.html#D10" target="_self" title="NET_1:def.10">Def10</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E4:43_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K10" title="NET_1:func.10">Ext</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="net_1.html#D15" target="_self" title="NET_1:def.15">Def15</a></span>;<br/>
<a NAME="E5:43_1_2"/>
<font color="Maroon" title="c3">x</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="c3">x</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/>
<span class="kw">hence </span><a NAME="E6:43_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_2"><span class="lab"><font color="Green" title="E27">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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="E3:43_1"/>
( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">hence </span><a NAME="E4:43_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E23">A2</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>
<span class="kw">assume </span><a NAME="E2:43"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
 <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> iff ( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) ) )</span><br/>

<a NAME="E3:43"/>
(  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) )</span><br/>

<a NAME="E2:43_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K10" title="NET_1:func.10">Ext</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:43_2"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K10" title="NET_1:func.10">Ext</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E5:43_2"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font><span class="kw"> such that </span><br/><a NAME="E7:43_2"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E8:43_2"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="net_1.html#K5" title="NET_1:func.5">exit</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c5">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="net_1.html#D15" target="_self" title="NET_1:def.15">Def15</a></span>;<br/>
<a NAME="E9:43_2"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> implies <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="net_1.html#K3" title="NET_1:func.3">Post</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c5">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="net_1.html#D10" target="_self" title="NET_1:def.10">Def10</a></span>;<br/>
<a NAME="E10:43_2"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_2_1"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</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">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">z</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="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">x</font> )</span><br/>

<a NAME="E2:43_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U3" title="NET_1:sel.3">Flow</a> of <font color="Maroon" title="c1">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E9:43_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E1:43_2_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="net_1.html#D8" target="_self" title="NET_1:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_2_1"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c3">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E1:43_2_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="net_1.html#D6" target="_self" title="NET_1:def.6">Def6</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="E11:43_2"/><span class="lab"><font color="Green" title="E30">A19</font></span>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E12:43_2"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> implies <font color="Maroon" title="c4">y</font> <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="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="net_1.html#D10" target="_self" title="NET_1:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:43_2"/>
( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E11:43_2"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E10:43_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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">hence </span><a NAME="E4:43"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K12" title="NET_1:func.12">Output</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">X</font> iff ( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U1" title="NET_1:sel.1">Places</a> of <font color="Maroon" title="c1">N</font> ) or  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="net_1.html#U2" title="NET_1:sel.2">Transitions</a> of <font color="Maroon" title="c1">N</font> &amp;  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c3">x</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E22">A1</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/>


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