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

<b>reconsider </b><font color="Maroon">V</font> = the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">E</font> = the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">Prod</font> = <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span> ex <font color="Olive">M</font> being  <a href="graph_1.html#NM1">Graph</a> ex <font color="Olive">v</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">e</font>,<font color="Olive">v</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K8">`1</a>  &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K9">`2</a>  &amp; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K10">`3</a>  &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K11">`4</a>  &amp; <font color="Olive">M</font> <a href="hidden.html#R1">=</a>  <a href="graph_1.html#G1" target="_self">MultiGraphStruct</a>(# <font color="Olive">v</font>,<font color="Olive">e</font>,<font color="Olive">s</font>,<font color="Olive">t</font> #) &amp; <font color="Olive">M</font> is    <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">G</font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span> ex <font color="Olive">M</font> being <a href="graph_1.html#V1" target="_self">strict</a> <a href="graph_1.html#NM1">Graph</a> ex <font color="Olive">v</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">e</font>,<font color="Olive">v</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K8">`1</a>  &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K9">`2</a>  &amp; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K10">`3</a>  &amp; <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="mcart_1.html#K11">`4</a>  &amp; <font color="Olive">M</font> <a href="hidden.html#R1">=</a>  <a href="graph_1.html#G1" target="_self">MultiGraphStruct</a>(# <font color="Olive">v</font>,<font color="Olive">e</font>,<font color="Olive">s</font>,<font color="Olive">t</font> #) &amp; <font color="Olive">M</font> is    <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">M</font> );<br/>
<a NAME="E4:41_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>] holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 
;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:41_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E7:41_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies <font color="Maroon">x</font> is  <a href="graph_1.html#V1" target="_self">strict</a>  <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:41_1_1_1"/><b>then </b>
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">z</font>,<font color="Maroon">x</font>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:41_1_1_1"/>
<font color="Maroon">x</font> is  <a href="graph_1.html#V1" target="_self">strict</a>  <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">G</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E8:41_1_1"/>
<font color="Maroon">x</font> is  <a href="graph_1.html#V1" target="_self">strict</a>  <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">x</font> as  <a href="graph_1.html#V1" target="_self">strict</a>  <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E10:41_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Maroon">x</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">y</font> = <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default">the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">H</font>,the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">H</font>,the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">H</font>,the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">H</font></span><a href="mcart_1.html#K4">]</a></span>;<br/>

<a NAME="E1:41_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_1_1_2_1"/><i><font color="Green">E37</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:41_1_1_2_1"/><i><font color="Green">E38</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:41_1_1_2_1"/>
( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:41_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span>
 <b>by </b><i>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="mcart_1.html#T84">MCART_1:84</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">v</font> = the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">H</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">e</font> = the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">H</font>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">H</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">H</font>,<font color="Maroon">v</font> ;<br/>
<b>reconsider </b><font color="Maroon">t</font> = the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">H</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">H</font>,<font color="Maroon">v</font> ;<br/>
<a NAME="E6:41_1_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="mcart_1.html#K8">`1</a>  &amp; the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="mcart_1.html#K9">`2</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="mcart_1.html#K10">`3</a>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="mcart_1.html#K11">`4</a>  )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T59">MCART_1:59</a></i>;<br/>
<b>hence </b><a NAME="E7:41_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:41_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:41_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div>
