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

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:7_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E1:7_1_1_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:7_1_1_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>8</sub></font> = the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>thus </b><a NAME="E2:7_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_1_1_1_1"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:7_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E3:7_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E4:7_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:7_1_1_1_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><a NAME="E6:7_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E2:7_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E5:7_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:7"><i><font color="Green">E4</font></i></a>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:7_1_1_1_1"/>
( not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_1_1_2"/><i><font color="Green">E8</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:7_1_1_1_1_2"/><b>then </b><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>8</sub></font> = the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>thus </b><a NAME="E3:7_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_1_1_2_1"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:7_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:7_1_1_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E4:7_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>]
 <b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_1_1_2"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="graph_1.html#E2:7_1_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:7_1_1_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="graph_1.html#E1:7_1_1_1"><i><font color="Green">E5</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E4:7_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:7_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:7_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="graph_1.html#E1:7_1_1_1_2"><i><font color="Green">E7</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:7_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E6:7_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:7_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="graph_1.html#E5:7_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:7_1_1"/><i><font color="Green">E5</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<a NAME="E4:7_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E1:7_1_1_2"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_2_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:7_1_1_2_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_2_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>9</sub></font> = the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>thus </b><a NAME="E2:7_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_2_1_1_1"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:7_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_2_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E3:7_1_1_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E4:7_1_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:7_1_1_2_1_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_2_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><a NAME="E6:7_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_2_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E2:7_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E5:7_1_1_2_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:7"><i><font color="Green">E4</font></i></a>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:7_1_1_2_1"/>
( not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1_1_2_1_2"/><i><font color="Green">E9</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:7_1_1_2_1_2"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_2_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>9</sub></font> = the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>thus </b><a NAME="E3:7_1_1_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_1_2_1_2_1"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:7_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:7_1_1_2_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E4:7_1_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>]
 <b>by </b><i><a class="txt" href="graph_1.html#E1:7_1_1_2_1_2"><i><font color="Green">E9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="graph_1.html#E2:7_1_1_2_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:7_1_1_2"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="graph_1.html#E1:7_1_1_2"><i><font color="Green">E6</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E4:7_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1_2_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:7_1_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:7_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="graph_1.html#E1:7_1_1_2_2"><i><font color="Green">E8</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:7_1_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E6:7_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:7_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="graph_1.html#E5:7_1_1_2"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E6:7_1_1"/><i><font color="Green">E6</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<a NAME="E7:7_1_1"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:7_1_1_3"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:7_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:7_1_1_3"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="graph_1.html#G1" target="_self">MultiGraphStruct</a>(# <span class="p1">(<span class="default">the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default">the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) as  <a href="graph_1.html#V1" target="_self">strict</a> <a href="graph_1.html#NM1">Graph</a> <b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E9:7_1_1"/>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><a NAME="E10:7_1_1"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><a NAME="E11:7_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E3:7_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="graph_1.html#E6:7_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E12:7_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>hence </b><a NAME="E13:7_1_1"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:7_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:7_1_1"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E6:7_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="graph_1.html#E12:7_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div>
