<?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>let </b><font color="Maroon">G3</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/>





<b>assume </b><a NAME="E1:47"/><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> &amp; 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">G3</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">G3</font> &amp; the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G2</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G3</font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G2</font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G3</font> )
 ;<br/>

<a NAME="E2:47"/><i><font color="Green">E35</font></i>: 
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> <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 <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span></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">G3</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 <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><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="E1:47_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">v</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 <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span></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">G3</font></span>)</span>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <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> &amp; <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 <font color="Maroon">G3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_1"/>
<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>
 
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_1"/><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/>;<br/>
<a NAME="E5:47_1"/><b>then </b><i><font color="Green">E38</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#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1"/><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>
 ;<br/><a NAME="E2:47_1_1"/><b>then </b>
<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 <font color="Maroon">G1</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">v</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">G1</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">G3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </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#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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#E2:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_2"/><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">G2</font>
 ;<br/><a NAME="E2:47_1_2"/><b>then </b>
<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 <font color="Maroon">G2</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">v</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">G2</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">G3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </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#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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#E2:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:47_1"/>
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 <font color="Maroon">G3</font> <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="E3:47"/>
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> <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 <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span></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">G3</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 <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><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="E1:47_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">v</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 <span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G2</font></span>)</span></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">G3</font></span>)</span>
 ;<br/>

<a NAME="E2:47_2"/><b>then </b><i><font color="Green">E45</font></i>: 
( <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> &amp; <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 <font color="Maroon">G3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_2"/>
<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>
 
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_2"/><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/>;<br/>
<a NAME="E5:47_2"/><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#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_2_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/><a NAME="E2:47_2_1"/><b>then </b>
<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 <font color="Maroon">G1</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">v</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">G1</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">G3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </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#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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#E2:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_2_2"/><i><font color="Green">E51</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/><a NAME="E2:47_2_2"/><b>then </b>
<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 <font color="Maroon">G2</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_2_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">v</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">G2</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">G3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </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#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</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#E2:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:47_2"/>
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 <font color="Maroon">G3</font> <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="E4:47"/><b>then </b><i><font color="Green">E53</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="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">G3</font> &amp; 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="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">G3</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green">E29</font></i></a>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green">E54</font></i>: 
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> <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">G1</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 <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> holds <br/>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> <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">G3</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="47_3"><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="E1:47_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">v</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">G1</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 <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:47_3"/><b>then </b><i><font color="Green">E56</font></i>: 
( <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 <font color="Maroon">G1</font> &amp; <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">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_3"/>
<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">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_3"/><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">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:47_3"/><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 <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">G3</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/>
<a NAME="E6:47_3"/><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 <font color="Maroon">G2</font> <a href="xboole_0.html#K2">\/</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">G3</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:47_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">v</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">G2</font></span>)</span> <a href="xboole_0.html#K2">\/</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">G3</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_3_1"/><i><font color="Green">E60</font></i>: 
<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 <font color="Maroon">G2</font>
 ;<br/><a NAME="E2:47_3_1"/><b>then </b><i><font color="Green">E61</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>
 <b>by </b><i/>;<br/><a NAME="E3:47_3_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">v</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">G1</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">G2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G3</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#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_3_2"/><i><font color="Green">E63</font></i>: 
<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 <font color="Maroon">G3</font>
 ;<br/><a NAME="E2:47_3_2"/><b>then </b><i><font color="Green">E64</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">G3</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_3_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">v</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">G1</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">G3</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G3</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">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:47_3"/>
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> <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">G3</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#E2:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:47"/>
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> <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">G1</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 <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> holds <br/>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> <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">G3</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="47_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="E1:47_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">v</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">G1</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 <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:47_4"/><b>then </b><i><font color="Green">E67</font></i>: 
( <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 <font color="Maroon">G1</font> &amp; <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">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_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#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">G3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_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">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:47_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 <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">G3</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/>
<a NAME="E6:47_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 <font color="Maroon">G2</font> <a href="xboole_0.html#K2">\/</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">G3</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:47_4"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">v</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">G2</font></span>)</span> <a href="xboole_0.html#K2">\/</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">G3</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_4_1"/><i><font color="Green">E70</font></i>: 
<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 <font color="Maroon">G2</font>
 ;<br/><a NAME="E2:47_4_1"/><b>then </b><i><font color="Green">E71</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>
 <b>by </b><i/>;<br/><a NAME="E3:47_4_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">v</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">G1</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">G2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G3</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#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_4_2"/><i><font color="Green">E130</font></i>: 
<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 <font color="Maroon">G3</font>
 ;<br/><a NAME="E2:47_4_2"/><b>then </b><i><font color="Green">E131</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">G3</font>
 <b>by </b><i/>;<br/><a NAME="E3:47_4_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">v</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">G1</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">G3</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G3</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">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:47_4"/>
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> <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">G3</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#E2:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:47"/><b>then </b><i><font color="Green">E133</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 <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> &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 <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</a></i>;<br/>
<i><font color="Green">E137</font></i>: the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> = 
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> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">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></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>
<br/>.= 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">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">G3</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
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 <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<i><font color="Green">E138</font></i>: the <a href="graph_1.html#U1" target="_self">Vertices</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> = 
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="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">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></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>
<br/>.= 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">G1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">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">G3</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
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 <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>
<br/>.= 
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> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<i><font color="Green">E139</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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> = 
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> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
<b>by </b><i>, </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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E11:47"/>
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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> holds <br/>the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_8"><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="E1:47_8"/><i><font color="Green">E140</font></i>: 
<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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
 ;<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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> = 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
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> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">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></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>
;<br/>
<a NAME="E3:47_8"/><b>then </b><i><font color="Green">E141</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> or <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">G3</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E142</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_8_2"/><i><font color="Green">E143</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/><a NAME="E2:47_8_2"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</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="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_8_2"/><b>then </b>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</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="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><b>hence </b>the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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 <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>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green">E144</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_8_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_8_3"/><i><font color="Green">E145</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/><a NAME="E2:47_8_3"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</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="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_8_3"/><b>then </b><i><font color="Green">E146</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</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="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><a NAME="E4:47_8_3"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a> 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">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:47_8_3"/><b>then </b><i><font color="Green">E147</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a> 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">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><b>thus </b>the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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 <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>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, , , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</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">G3</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#E4"><i><font color="Green">Lemma28</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_8_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_8_4"/><i><font color="Green">E148</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">G3</font>
 ;<br/><a NAME="E2:47_8_4"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font> <a href="tarski.html#R1">c=</a> 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">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_8_4"/><b>then </b><i><font color="Green">E149</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font> <a href="tarski.html#R1">c=</a> 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">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><b>thus </b>the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</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">G3</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#E4"><i><font color="Green">Lemma28</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:47_8"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></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="E12:47"/><b>then </b><i><font color="Green">E150</font></i>: 
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<i><font color="Green">E151</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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> = 
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> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
<b>by </b><i>, </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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E14:47"/>
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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> holds <br/>the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_10"><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="E1:47_10"/><i><font color="Green">E152</font></i>: 
<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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
 ;<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"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span> = 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
<b>by </b><i/>
<br/>.= 
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> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default">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></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>
;<br/>
<a NAME="E3:47_10"/><b>then </b><i><font color="Green">E153</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> or <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">G3</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_10_2"/><i><font color="Green">E155</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/><a NAME="E2:47_10_2"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</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="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_10_2"/><b>then </b>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G1</font> <a href="tarski.html#R1">c=</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="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><b>hence </b>the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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 <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>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green">E156</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_10_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_10_3"/><i><font color="Green">E157</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/><a NAME="E2:47_10_3"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</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="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_10_3"/><b>then </b><i><font color="Green">E158</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</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="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><a NAME="E4:47_10_3"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a> 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">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:47_10_3"/><b>then </b><i><font color="Green">E159</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G2</font> <a href="tarski.html#R1">c=</a> 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">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><b>thus </b>the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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 <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>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a>, , , </i>
<br/>.= 
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#E4"><i><font color="Green">Lemma28</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">G3</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#E4"><i><font color="Green">Lemma28</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_10_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_10_4"/><i><font color="Green">E160</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">G3</font>
 ;<br/><a NAME="E2:47_10_4"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font> <a href="tarski.html#R1">c=</a> 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">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_10_4"/><b>then </b><i><font color="Green">E161</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">G3</font> <a href="tarski.html#R1">c=</a> 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">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><b>thus </b>the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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">G3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</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">G3</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#E4"><i><font color="Green">Lemma28</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:47_10"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</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">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span></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>
<b>hence </b><a NAME="E15:47"/>
<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="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="graph_1.html#K1" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">G3</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
