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<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="E1:60"/><i><font color="Green">E38</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>assume </b><a NAME="E2:60"/><i><font color="Green">E39</font></i>: 
( <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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:60"/><b>then </b>
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>4</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>3</sub></font> <a href="xboole_0.html#K2">\/</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="txt" href="graph_1.html#E1:60"><i><font color="Green">E38</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:60"/><b>then </b><i><font color="Green">E40</font></i>: 
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>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:60"/>
the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>4</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>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#E1:60"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_1.html#E2:60"><i><font color="Green">E39</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:60"/><b>then </b><i><font color="Green">E41</font></i>: 
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="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:60"/>
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#U2" target="_self">Edges</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> &amp; 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> &amp; 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> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> &amp; 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> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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="E1:60_1"/><i><font color="Green">E42</font></i>: 
<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>
 ;<br/>

<b>hence </b><a NAME="E2:60_1"/><i><font color="Green">E43</font></i>: 
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#E1:60"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_1.html#E2:60"><i><font color="Green">E39</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/>

<b>thus </b><a NAME="E3:60_1"/><i><font color="Green">E44</font></i>: 
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#E1:60"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_1.html#E2:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_1.html#E1:60_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/>

<b>thus </b><a NAME="E4:60_1"/>
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> <a href="hidden.html#R2">in</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="txt" href="graph_1.html#E1:60_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_1.html#E2:60_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="graph_1.html#E22"><i><font color="Green">Lemma17</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:60_1"/>
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> <a href="hidden.html#R2">in</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="txt" href="graph_1.html#E1:60_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_1.html#E3:60_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="graph_1.html#E22"><i><font color="Green">Lemma17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:60"/><b>then </b>
<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="txt" href="graph_1.html#E4:60"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_1.html#E6:60"><i><font color="Green">E41</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E9:60"/>
<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>
 <b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self">Def23</a></i>;<br/>


</div>
