<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:35"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="sgraph1.html#R2" target="_self">is_SetOfSimpleGraphs_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p2">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> #)]
 
<b>by </b><i><a class="ref" href="sgraph1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</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="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p2">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> #)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:35_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>3</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sgraph1.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E3:35_1"/>
 <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <span class="p2">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> #) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:35"><i><font color="Green">E21</font></i></a>, <a class="txt" href="sgraph1.html#E1:35_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="sgraph1.html#D9" target="_self">Def9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:35"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> #)] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E2:35_2"/><i><font color="Green">E26</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>3</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sgraph1.html#T27" target="_self">Th27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:35_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:35_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="sgraph1.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:35_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> #) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:35"><i><font color="Green">E21</font></i></a>, <a class="txt" href="sgraph1.html#E1:35_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="sgraph1.html#E2:35_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="sgraph1.html#E4:35_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sgraph1.html#D9" target="_self">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E7:35_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font> #) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E4:35_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="sgraph1.html#E6:35_2"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:35"/><i><font color="Green">E25</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="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="sgraph1.html#S1" target="_self">SGRAPH1:sch 1</a>(<a class="txt" href="sgraph1.html#E2:35"><i><font color="Green">E22</font></i></a>, <a class="txt" href="sgraph1.html#E3:35"><i><font color="Green">E23</font></i></a>, <a class="txt" href="sgraph1.html#E4:35"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:35"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E7:35"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E5:35"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
