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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/>





<b>assume </b><a NAME="E1:53"/>
( <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>5</sub></font> )
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green">E32</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>5</sub></font> )
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E3:53"/><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>5</sub></font> )
 
<b>by </b><i><a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green">E33</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>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green">E34</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> &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>5</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:53"><i><font color="Green">E32</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<a NAME="E6:53"/><b>then </b><i><font color="Green">E35</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>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:53"/>
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>5</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>5</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>5</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>5</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="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E36</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>
 ;<br/>

<b>hence </b><i><font color="Green">E37</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>6</sub></font> = 
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#E2:53"><i><font color="Green">E32</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>
<br/>.= 
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</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#E2:53"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_1.html#E5:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>
;<br/><br/>

<b>thus </b><i><font color="Green">E38</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>6</sub></font> = 
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#E2:53"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>
<br/>.= 
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</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#E2:53"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_1.html#E5:53"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>
;<br/>

<a NAME="E4:53_1"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> 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#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:53_1"/>
<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>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E6:53_1"/><b>then </b>
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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:53_1"/>
the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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#E2:53_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_1.html#E4:53_1"><i><font color="Green">E39</font></i></a></i>;<br/>

<a NAME="E8:53_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> 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#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:53_1"/>
<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>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E10:53_1"/><b>then </b>
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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:53_1"/>
the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</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#E3:53_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_1.html#E8:53_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:53"/><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>5</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4:53"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_1.html#E6:53"><i><font color="Green">E35</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self">Def17</a></i>;<br/>
<b>hence </b><a NAME="E9:53"/>
<font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#R4" target="_self">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self">Def23</a></i>;<br/>


</div>
