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

<b>let </b><font color="Maroon">G</font> be  <a href="graph_1.html#V1" target="_self">strict</a> <a href="graph_1.html#NM1">Graph</a>, <font color="Maroon">G'</font> be  <a href="graph_1.html#V1" target="_self">strict</a> <a href="graph_1.html#NM1">Graph</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7_1_2"/><i><font color="Green">E60</font></i>: 
( the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G2</font> &amp; the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> &amp; ( for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> holds <br/>( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> ) ) &amp; ( for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> holds <br/>( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> ) ) )
 <b>and </b><br/><a NAME="E2:7_1_2"/><i><font color="Green">E61</font></i>: 
( the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G'</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G2</font> &amp; the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G'</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> &amp; ( for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> holds <br/>( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> ) ) &amp; ( for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> holds <br/>( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> ) ) )
 ;<br/>

<a NAME="E3:7_1_2"/><i><font color="Green">E62</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:7_1_2"/><i><font color="Green">E63</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G'</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:7_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> holds <br/>the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:7_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>
 ;<br/>

<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font>
 ;<br/><b>hence </b>the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font>
 ;<br/><b>hence </b>the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_2_1"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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>
<a NAME="E6:7_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G'</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E7:7_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font> holds <br/>the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:7_1_2_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G</font>
 ;<br/>

<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font>
 ;<br/><b>hence </b>the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2_2_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font>
 ;<br/><b>hence </b>the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7_1_2_2"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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>hence </b><a NAME="E8:7_1_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G'</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
