<?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="E1:69"/><i><font color="Green">E39</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/>

<a NAME="E2:69"/><i><font color="Green">E40</font></i>: 
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="graph_1.html#K7" target="_self">bool</a> <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>  <a href="graph_1.html#K7" target="_self">bool</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_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="E1:69_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="graph_1.html#V1" target="_self">strict</a>  <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#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E3:69_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="graph_1.html#R4" target="_self">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="graph_1.html#R4" target="_self">c=</a> <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> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="graph_1.html#D23" target="_self">Def23</a>, <a class="ref" href="graph_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E4:69_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="graph_1.html#R4" target="_self">c=</a> <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>
 
<b>by </b><i><a class="ref" href="graph_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E5:69_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is  <a href="graph_1.html#V1" target="_self">strict</a>  <a href="graph_1.html#M3" target="_self">Subgraph</a> of <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>
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self">Def23</a></i>;<br/>
<b>hence </b><a NAME="E6:69_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <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="ref" href="graph_1.html#D24" target="_self">Def24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:69"/><i><font color="Green">E41</font></i>: 
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="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_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="E1:69_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="graph_1.html#V1" target="_self">strict</a>  <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#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E3:69_2"/>
( <font color="Maroon">c<sub>6</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> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:69"><i><font color="Green">E39</font></i></a>, <a class="ref" href="graph_1.html#D23" target="_self">Def23</a>, <a class="ref" href="graph_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E4:69_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="graph_1.html#R4" target="_self">c=</a> <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>
 
<b>by </b><i><a class="ref" href="graph_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E5:69_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is  <a href="graph_1.html#V1" target="_self">strict</a>  <a href="graph_1.html#M3" target="_self">Subgraph</a> of <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>
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self">Def23</a></i>;<br/>
<b>hence </b><a NAME="E6:69_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <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="ref" href="graph_1.html#D24" target="_self">Def24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:69"/>
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> <span class="p1">(<span class="default"><a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><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="E1:69_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:69_3"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <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/>
<b>hence </b><a NAME="E3:69_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="graph_1.html#K7" target="_self">bool</a> <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#E2:69"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_1.html#E3:69"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:69"/>
<span class="p1">(<span class="default"><a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" target="_self">bool</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="graph_1.html#K7" target="_self">bool</a> <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="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div>
