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

<b>let </b><font color="Maroon" title="c3">G1</font>, <font color="Maroon" title="c4">G2</font>, <font color="Maroon" title="c5">G3</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E28">A1</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> )
 ;<br/>

<a NAME="E2:47"/><i><font color="Green" title="E29">A2</font></i>: 
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> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font></span>)</span> holds <br/>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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <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 <font color="Maroon" title="c5">G3</font> <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="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_1"/><b>then </b>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</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="E5:47_1"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c6">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">G1</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="c4">G2</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/>
<div><i><font color="Green" title="E33">A6</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" title="E34">A7</font></i>: 
<font color="Maroon" title="c6">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">G1</font>
 ;<br/><a NAME="E2:47_1_1"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</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/><a NAME="E3:47_1_1"/><b>then </b><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="E34">A9</font></i>: 
<font color="Maroon" title="c6">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="c4">G2</font>
 ;<br/><a NAME="E2:47_1_2"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</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/><a NAME="E3:47_1_2"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_1_2"><i><font color="Green" title="E34">A9</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_1_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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 <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E5:47_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="graph_1.html#E6:47_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47"/>
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> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font></span>)</span> holds <br/>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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <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 <font color="Maroon" title="c5">G3</font> <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="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_2"/><i><font color="Green" title="E30">A11</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 ;<br/>

<a NAME="E2:47_2"/><b>then </b><i><font color="Green" title="E31">A12</font></i>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_2"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47_2"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_2"/><b>then </b>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</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="E5:47_2"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon" title="c6">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">G1</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="c4">G2</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/>
<div><i><font color="Green" title="E33">A14</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" title="E34">A15</font></i>: 
<font color="Maroon" title="c6">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">G1</font>
 ;<br/><a NAME="E2:47_2_1"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</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/><a NAME="E3:47_2_1"/><b>then </b><i><font color="Green" title="E35">A16</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_2_1"><i><font color="Green" title="E34">A15</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_2_1"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="E34">A17</font></i>: 
<font color="Maroon" title="c6">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="c4">G2</font>
 ;<br/><a NAME="E2:47_2_2"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</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/><a NAME="E3:47_2_2"/><b>then </b><i><font color="Green" title="E35">A18</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_2_2"><i><font color="Green" title="E34">A17</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_2_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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 <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E5:47_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="graph_1.html#E6:47_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/><b>then </b><i><font color="Green" title="E30">A19</font></i>: 
( 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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> &amp; 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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:47"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green" title="E31">A20</font></i>: 
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> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> holds <br/>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</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="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_3"/><i><font color="Green" title="E32">A21</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:47_3"/><b>then </b><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_3"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47_3"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_3"/><b>then </b>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</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="E5:47_3"/><b>then </b>
<font color="Maroon" title="c6">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="c4">G2</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="c5">G3</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/>
<a NAME="E6:47_3"/><b>then </b>
<font color="Maroon" title="c6">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="c4">G2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</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="E7:47_3"/><b>then </b><i><font color="Green" title="E34">A23</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</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/>
<div><i><font color="Green" title="E35">A24</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" title="E36">A25</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font>
 ;<br/><a NAME="E2:47_3_1"/><b>then </b><i><font color="Green" title="E37">A26</font></i>: 
<font color="Maroon" title="c6">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="c4">G2</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/><a NAME="E3:47_3_1"/><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_3"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="graph_1.html#E1:47_3_1"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E2:47_3_1"><i><font color="Green" title="E37">A26</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_3_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="E36">A28</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font>
 ;<br/><a NAME="E2:47_3_2"/><b>then </b><i><font color="Green" title="E37">A29</font></i>: 
<font color="Maroon" title="c6">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="c5">G3</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/><a NAME="E3:47_3_2"/><i><font color="Green" title="E38">A30</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_3"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="graph_1.html#E1:47_3_2"><i><font color="Green" title="E36">A28</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E2:47_3_2"><i><font color="Green" title="E37">A29</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_3_2"><i><font color="Green" title="E38">A30</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E7:47_3"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="graph_1.html#E8:47_3"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:47"/>
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> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> holds <br/>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</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="47_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_4"/><i><font color="Green" title="E32">A31</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:47_4"/><b>then </b><i><font color="Green" title="E33">A32</font></i>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:47_4"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47_4"><i><font color="Green" title="E32">A31</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:47_4"/><b>then </b>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</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="E5:47_4"/><b>then </b>
<font color="Maroon" title="c6">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="c4">G2</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="c5">G3</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/>
<a NAME="E6:47_4"/><b>then </b>
<font color="Maroon" title="c6">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="c4">G2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</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="E7:47_4"/><b>then </b><i><font color="Green" title="E34">A33</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</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/>
<div><i><font color="Green" title="E35">A34</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" title="E36">A35</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font>
 ;<br/><a NAME="E2:47_4_1"/><b>then </b><i><font color="Green" title="E37">A36</font></i>: 
<font color="Maroon" title="c6">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="c4">G2</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/><a NAME="E3:47_4_1"/><i><font color="Green" title="E38">A37</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_4"><i><font color="Green" title="E33">A32</font></i></a>, <a class="txt" href="graph_1.html#E1:47_4_1"><i><font color="Green" title="E36">A35</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E2:47_4_1"><i><font color="Green" title="E37">A36</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_4_1"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="E36">A38</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font>
 ;<br/><a NAME="E2:47_4_2"/><b>then </b><i><font color="Green" title="E37">A39</font></i>: 
<font color="Maroon" title="c6">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="c5">G3</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/><a NAME="E3:47_4_2"/><i><font color="Green" title="E38">A40</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:47_4"><i><font color="Green" title="E33">A32</font></i></a>, <a class="txt" href="graph_1.html#E1:47_4_2"><i><font color="Green" title="E36">A38</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E2:47_4_2"><i><font color="Green" title="E37">A39</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E3:47_4_2"><i><font color="Green" title="E38">A40</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">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" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E7:47_4"><i><font color="Green" title="E34">A33</font></i></a>, <a class="txt" href="graph_1.html#E8:47_4"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:47"/><b>then </b><i><font color="Green" title="E32">A41</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E5:47"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></i>;<br/>
<i><font color="Green" title="E33">A42</font></i>: the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <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">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</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="c5">G3</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
;<br/>
<i><font color="Green" title="E34">A43</font></i>: the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <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">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default">the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c4">G2</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="c5">G3</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</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 <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
;<br/>
<i><font color="Green" title="E35">A44</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E8:47"><i><font color="Green" title="E33">A42</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#NK2" title="RELAT_1:NK.2">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></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="E11:47"/>
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#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> holds <br/>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></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="47_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_8"/><i><font color="Green" title="E36">A45</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 ;<br/>

 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</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/>.= 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
;<br/>
<a NAME="E3:47_8"/><b>then </b><i><font color="Green" title="E37">A46</font></i>: 
( <font color="Maroon" title="c6">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">G1</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="c4">G2</font> or <font color="Maroon" title="c6">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="c5">G3</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47_8"><i><font color="Green" title="E36">A45</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green" title="E38">A47</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" title="E39">A48</font></i>: 
<font color="Maroon" title="c6">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">G1</font>
 ;<br/><a NAME="E2:47_8_2"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_8_2"/><b>then </b>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><b>hence </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_2"><i><font color="Green" title="E39">A48</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_2"><i><font color="Green" title="E39">A48</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_2"><i><font color="Green" title="E39">A48</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E39">A49</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" title="E40">A50</font></i>: 
<font color="Maroon" title="c6">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="c4">G2</font>
 ;<br/><a NAME="E2:47_8_3"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_8_3"/><b>then </b><i><font color="Green" title="E41">A51</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><a NAME="E4:47_8_3"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</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="c5">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:47_8_3"/><b>then </b><i><font color="Green" title="E42">A52</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><b>thus </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_3"><i><font color="Green" title="E40">A50</font></i></a>, <a class="txt" href="graph_1.html#E3:47_8_3"><i><font color="Green" title="E41">A51</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_3"><i><font color="Green" title="E40">A50</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_3"><i><font color="Green" title="E40">A50</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_3"><i><font color="Green" title="E40">A50</font></i></a>, <a class="txt" href="graph_1.html#E5:47_8_3"><i><font color="Green" title="E42">A52</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></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" title="E40">A53</font></i>: 
<font color="Maroon" title="c6">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="c5">G3</font>
 ;<br/><a NAME="E2:47_8_4"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</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="c5">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_8_4"/><b>then </b><i><font color="Green" title="E41">A54</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><b>thus </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_4"><i><font color="Green" title="E40">A53</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_4"><i><font color="Green" title="E40">A53</font></i></a>, <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#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="txt" href="graph_1.html#E1:47_8_4"><i><font color="Green" title="E40">A53</font></i></a>, <a class="txt" href="graph_1.html#E3:47_8_4"><i><font color="Green" title="E41">A54</font></i></a>, <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="E7:47_8"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:47_8"><i><font color="Green" title="E37">A46</font></i></a>, <a class="txt" href="graph_1.html#E4:47_8"><i><font color="Green" title="E38">A47</font></i></a>, <a class="txt" href="graph_1.html#E5:47_8"><i><font color="Green" title="E39">A49</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:47"/><b>then </b><i><font color="Green" title="E36">A55</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E10:47"><i><font color="Green" title="E35">A44</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="E37">A56</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_1.html#E8:47"><i><font color="Green" title="E33">A42</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#NK2" title="RELAT_1:NK.2">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></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="E14:47"/>
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#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> holds <br/>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></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="47_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:47_10"/><i><font color="Green" title="E38">A57</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 ;<br/>

 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> = 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</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/>.= 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font></span>)</span> <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="c5">G3</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
;<br/>
<a NAME="E3:47_10"/><b>then </b><i><font color="Green" title="E39">A58</font></i>: 
( <font color="Maroon" title="c6">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">G1</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="c4">G2</font> or <font color="Maroon" title="c6">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="c5">G3</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:47_10"><i><font color="Green" title="E38">A57</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green" title="E40">A59</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" title="E41">A60</font></i>: 
<font color="Maroon" title="c6">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">G1</font>
 ;<br/><a NAME="E2:47_10_2"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_10_2"/><b>then </b>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><b>hence </b>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_2"><i><font color="Green" title="E41">A60</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_2"><i><font color="Green" title="E41">A60</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_2"><i><font color="Green" title="E41">A60</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E41">A61</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" title="E42">A62</font></i>: 
<font color="Maroon" title="c6">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="c4">G2</font>
 ;<br/><a NAME="E2:47_10_3"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_10_3"/><b>then </b><i><font color="Green" title="E43">A63</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><a NAME="E4:47_10_3"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</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="c5">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:47_10_3"/><b>then </b><i><font color="Green" title="E44">A64</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><b>thus </b>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_3"><i><font color="Green" title="E42">A62</font></i></a>, <a class="txt" href="graph_1.html#E3:47_10_3"><i><font color="Green" title="E43">A63</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_3"><i><font color="Green" title="E42">A62</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_3"><i><font color="Green" title="E42">A62</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_3"><i><font color="Green" title="E42">A62</font></i></a>, <a class="txt" href="graph_1.html#E5:47_10_3"><i><font color="Green" title="E44">A64</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></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" title="E42">A65</font></i>: 
<font color="Maroon" title="c6">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="c5">G3</font>
 ;<br/><a NAME="E2:47_10_4"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</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="c5">G3</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:47_10_4"/><b>then </b><i><font color="Green" title="E43">A66</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/><b>thus </b>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E4:47"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_4"><i><font color="Green" title="E42">A65</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_4"><i><font color="Green" title="E42">A65</font></i></a>, <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#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E7:47"><i><font color="Green" title="E32">A41</font></i></a>, <a class="txt" href="graph_1.html#E1:47_10_4"><i><font color="Green" title="E42">A65</font></i></a>, <a class="txt" href="graph_1.html#E3:47_10_4"><i><font color="Green" title="E43">A66</font></i></a>, <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="E7:47_10"/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:47_10"><i><font color="Green" title="E39">A58</font></i></a>, <a class="txt" href="graph_1.html#E4:47_10"><i><font color="Green" title="E40">A59</font></i></a>, <a class="txt" href="graph_1.html#E5:47_10"><i><font color="Green" title="E41">A61</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G2</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c5">G3</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E8:47"><i><font color="Green" title="E33">A42</font></i></a>, <a class="txt" href="graph_1.html#E9:47"><i><font color="Green" title="E34">A43</font></i></a>, <a class="txt" href="graph_1.html#E12:47"><i><font color="Green" title="E36">A55</font></i></a>, <a class="txt" href="graph_1.html#E13:47"><i><font color="Green" title="E37">A56</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
