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<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#V1" target="_self">strict</a> <a href="graph_1.html#NM1">Graph</a>;<br/>

<b>assume </b><a NAME="E1:52"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#R4" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="graph_1.html#R4" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:52"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="graph_1.html#M3" target="_self">Subgraph</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E3:52"/><b>then </b>
( the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E4:52"/><b>then </b><i><font color="Green">E32</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E5:52"/>
( the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> &amp; the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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#E2:52"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E6:52"/><b>then </b><i><font color="Green">E33</font></i>: 
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</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#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then </b><i><font color="Green">E34</font></i>:  <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> = 
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 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/>
<a NAME="E8:52"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <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="Olive">b<sub>1</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="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">c<sub>5</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>
 ;<br/>

<a NAME="E2:52_2"/><b>then </b>
<font color="Maroon">c<sub>5</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/>
<b>hence </b><a NAME="E3:52_2"/>
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>5</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>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:52"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:52"/><b>then </b><i><font color="Green">E35</font></i>: 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</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>
 
<b>by </b><i><a class="txt" href="graph_1.html#E7:52"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<i><font color="Green">E36</font></i>:  <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> = 
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#E6:52"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>
<br/>.= 
 <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/>
<a NAME="E11:52"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <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="Olive">b<sub>1</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="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:52_4"/>
<font color="Maroon">c<sub>5</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>
 ;<br/>

<a NAME="E2:52_4"/><b>then </b>
<font color="Maroon">c<sub>5</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/>
<b>hence </b><a NAME="E3:52_4"/>
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>5</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>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:52"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:52"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_1.html#E4:52"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_1.html#E6:52"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_1.html#E9:52"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_1.html#E10:52"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
