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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#NM1">Graph</a>;<br/>



<b>assume </b><a NAME="E1:43"/>
<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>
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E25</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>
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E3:43"/>
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> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> 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 <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><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="E1:43_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</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>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:43_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>6</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>7</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>6</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:43_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:43_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="E5:43_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</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="E6:43_1"/>
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>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:43"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E7:43_1"/><b>then </b>
<font color="Maroon">c<sub>6</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#E5:43_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E8:43_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</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/>
<a NAME="E9:43_1"/>
<font color="Maroon">c<sub>6</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#E4:43_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E10:43_1"/><b>then </b>
<font color="Maroon">c<sub>7</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>6</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:43"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E4:43_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E11:43_1"/>
<font color="Maroon">c<sub>5</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>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:43_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_1.html#E8:43_1"><i><font color="Green">E30</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="E4:43"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</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#D3">TARSKI:def 3</a></i>;<br/>

<a NAME="E5:43"/>
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> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> 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 <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_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="E1:43_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</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>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_2"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:43_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>6</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>7</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>6</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:43_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_1.html#E3:43_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E5:43_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</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="E6:43_2"/>
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>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:43"><i><font color="Green">E25</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E7:43_2"/><b>then </b>
<font color="Maroon">c<sub>6</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#E5:43_2"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E8:43_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</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/>
<a NAME="E9:43_2"/>
<font color="Maroon">c<sub>6</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#E4:43_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E10:43_2"/><b>then </b>
<font color="Maroon">c<sub>7</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>6</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:43"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_1.html#E4:43_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E11:43_2"/>
<font color="Maroon">c<sub>5</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>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:43_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_1.html#E8:43_2"><i><font color="Green">E30</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="E6:43"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</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#D3">TARSKI:def 3</a></i>;<br/>


</div>
