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

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="net_1.html#NM1" title="NET_1:NM.1">Pnet</a><br/> for <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="Olive" title="b1">N</font>  st <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="Olive" title="b1">N</font> holds <br/>( <font color="Maroon" title="c1">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="Olive" title="b1">N</font>,<font color="Olive" title="b2">y</font> iff  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Olive" title="b1">N</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c1">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">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="c2">N</font>  st <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="c2">N</font> holds <br/>( <font color="Maroon" title="c1">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="c2">N</font>,<font color="Olive" title="b1">y</font> iff  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c1">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</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="c2">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">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="c2">N</font> implies ( <font color="Maroon" title="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</font> iff  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">x</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c3">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="c2">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="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</font> iff  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">x</font> )</span><br/>

<a NAME="E2:27"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
(  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">x</font> implies <font color="Maroon" title="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:27_1"/>
 <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">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="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</font></span><br/>

<a NAME="E2:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">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="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="net_1.html#D6" target="_self" title="NET_1:def.6">Def6</a></span>;<br/>
<a NAME="E3:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">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="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="net_1.html#T12" target="_self" title="NET_1:th.12">Th12</a></span>;<br/>
<a NAME="E4:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:27_1"/>
<font color="Maroon" title="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="net_1.html#D8" target="_self" title="NET_1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:27"/>
( <font color="Maroon" title="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</font> implies  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:27_2"/>
<font color="Maroon" title="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</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#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">x</font></span><br/>

<a NAME="E2:27_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">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="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><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="E3:27_2"/>
 <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E12">A1</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>
<span class="kw">hence </span><a NAME="E4:27"/>
( <font color="Maroon" title="c1">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="c2">N</font>,<font color="Maroon" title="c3">y</font> iff  <a href="net_1.html#R2" title="NET_1:pred.2">post</a> <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E13">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>
