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

<b>let </b><font color="Maroon">G1</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green">E29</font></i>: 
( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G2</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G2</font> )
 ;<br/>

<b>then </b><i><font color="Green">E35</font></i>: the <a href="graph_1.html#U1" target="_self">Vertices</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> = 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G2</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G1</font>
<b>by </b><i/>
<br/>.= 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>
;<br/>
<i><font color="Green">E36</font></i>: the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> = 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>
<br/>.= 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>
;<br/>
<a NAME="E4:46"/><i><font color="Green">E37</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:46"/><i><font color="Green">E38</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<i><font color="Green">E39</font></i>:  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> = 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E1:46"><i><font color="Green">E29</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E7:46"/>
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>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> holds <br/>the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_4"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span>
 ;<br/>

<a NAME="E2:46_4"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:46_4"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">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> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_4_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">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>
 ;<br/><b>hence </b>the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>
<br/>.= 
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="graph_1.html#E2:46"><i><font color="Green">E35</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_4_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">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>
 ;<br/><b>hence </b>the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>
<br/>.= 
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="graph_1.html#E2:46"><i><font color="Green">E35</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:46_4"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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="E8:46"/><b>then </b><i><font color="Green">E44</font></i>: 
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<i><font color="Green">E45</font></i>:  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> = 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E1:46"><i><font color="Green">E29</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E10:46"/>
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>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> holds <br/>the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_6"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span>
 ;<br/>

<a NAME="E2:46_6"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:46_6"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">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> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_6_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">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>
 ;<br/><b>hence </b>the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>
<br/>.= 
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="graph_1.html#E4:46"><i><font color="Green">E37</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_6_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">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>
 ;<br/><b>hence </b>the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>
<br/>.= 
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="graph_1.html#E4:46"><i><font color="Green">E37</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:46_6"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_1.html#E3:46"><i><font color="Green">E36</font></i></a>, <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="E11:46"/>
<font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="graph_1.html#E1:46"><i><font color="Green">E29</font></i></a>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
