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

<b>let </b><font color="Maroon" title="c3">G</font> be  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/>

<i><font color="Green" title="E26">A1</font></i>: the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> = 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
<br/>.= 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G</font>

;<br/>
<i><font color="Green" title="E27">A2</font></i>: the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font>
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
<br/>.= 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font>

;<br/>
<b>then </b><i><font color="Green" title="E28">A3</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>
;<br/>
<a NAME="E4:45"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> holds <br/>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_4"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font>
 ;<br/>

<a NAME="E2:45_4"/><b>then </b>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:45_4"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:45"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3:45"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<i><font color="Green" title="E30">A5</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E2:45"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>
;<br/>
<a NAME="E7:45"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> holds <br/>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_6"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font>
 ;<br/>

<a NAME="E2:45_6"/><b>then </b>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:45_6"/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:45"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:45"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="graph_1.html#E2:45"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="graph_1.html#E5:45"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="graph_1.html#E6:45"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
