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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green">E25</font></i>: 
( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E2:44"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:44_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> implies <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_1_1"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:44_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:44_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E5:44_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E6:44_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:44_1_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E1:44_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/><a NAME="E4:44_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E2:44_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:44_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E1:44_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/><a NAME="E4:44_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E2:44_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:44_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:44_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_1.html#E6:44_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E7:44_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:44_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_1_2"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:44_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E3:44_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E5:44_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E6:44_1_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E4:44_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E7:44_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:44_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E9:44_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E10:44_1_2"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E4:44_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E5:44_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E11:44_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:44_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_1.html#E9:44_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_1_3"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:44_1_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E3:44_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E5:44_1_3"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E6:44_1_3"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E4:44_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E7:44_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:44_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E9:44_1_3"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E10:44_1_3"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E4:44_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E5:44_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E11:44_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:44_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_1.html#E9:44_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:44_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:44"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<a NAME="E4:44"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:44_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> implies <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_2_1"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:44_2_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:44_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E5:44_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E6:44_2_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:44_2_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E1:44_2_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/><a NAME="E4:44_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E2:44_2_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:44_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:44_2_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E1:44_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="graph_1.html#E1:44_2_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/><a NAME="E4:44_2_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E2:44_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:44_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:44_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:44_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_1.html#E6:44_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E7:44_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:44_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_2_2"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:44_2_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44_2_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E3:44_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E5:44_2_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E6:44_2_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E4:44_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E7:44_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:44_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E9:44_2_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E10:44_2_2"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E4:44_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E5:44_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E11:44_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:44_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_1.html#E9:44_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_2_3"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:44_2_3"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E3:44_2_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E5:44_2_3"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E6:44_2_3"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_1.html#E4:44_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E7:44_2_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:44_2_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E9:44_2_3"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E10:44_2_3"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E4:44_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_1.html#E5:44_2_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E11:44_2_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:44_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_1.html#E9:44_2_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:44_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:44_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:44"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
