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

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

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c<sub>3</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> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> #)
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="sgraph1.html#E1:29_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="sgraph1.html#E3:29_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="sgraph1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E5:29_1"/><i><font color="Green">E21</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</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>4</sub></font> )
 
<b>by </b><i><a class="ref" href="sgraph1.html#T23" target="_self">Th23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1"/><b>then </b>
the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>4</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="sgraph1.html#E5:29_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="sgraph1.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:29"><i><font color="Green">E16</font></i></a>, <a class="txt" href="sgraph1.html#E5:29_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="sgraph1.html#E1:29_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/>
not <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">F<sub>1</sub></font>() as  non <a href="xboole_0.html#V1">empty</a>  <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">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">a<sub>1</sub></font>,<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> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> #)];<br/><a NAME="E3:29_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:29"><i><font color="Green">E16</font></i></a></i>;<br/><div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>assume </b><a NAME="E1:29_1_1_2_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>] &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<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> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> #);<br/><a NAME="E2:29_1_1_2_1"/>
<font color="Maroon">c<sub>6</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>5</sub></font>
 <b>by </b><i><a class="ref" href="finsub_1.html#T32">FINSUB_1:32</a></i>;<br/><a NAME="E3:29_1_1_2_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<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> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> #) <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<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> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> #) as    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="sgraph1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E5:29_1_1_2_1"/>
<font color="Maroon">P<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="Maroon">c<sub>9</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>7</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>9</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>7</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> #)]
 <b>by </b><i><a class="txt" href="sgraph1.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="sgraph1.html#E1:29_1_1_2_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="sgraph1.html#E3:29_1_1_2_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E6:29_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</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>7</sub></font></span><a href="tarski.html#K1">}</a></span>]
 ;<br/></div><b>end;</b></div><a NAME="E5:29_1_1_2"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>5</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S2">SETWISEO:sch 2</a>(<a class="txt" href="sgraph1.html#E3:29_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="sgraph1.html#E4:29_1_1_2"><i><font color="Green">E23</font></i></a>);<br/></i><div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>5</sub></font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_2_1"/>
(  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E2:29_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E3:29_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:29_1_1_2_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_2_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)]
 <b>by </b><i><a class="txt" href="sgraph1.html#E5:29_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_2"/>
 <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">c<sub>7</sub></font>] means for <font color="Olive">b<sub>1</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> <font color="Maroon">c<sub>6</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> #)];<br/><a NAME="E3:29_1_1_2_2_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">c<sub>7</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:29_1_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:29_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:29_1_1_2_2_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)]
 <b>by </b><i><a class="txt" href="sgraph1.html#E5:29_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>assume </b><a NAME="E1:29_1_1_2_2_1_2_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font>] &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finsub_1.html#T32">FINSUB_1:32</a></i>;<br/><a NAME="E3:29_1_1_2_2_1_2_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="finsub_1.html#T32">FINSUB_1:32</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #);<br/><a NAME="E4:29_1_1_2_2_1_2_2"/>
 <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) is    <a href="subset_1.html#M1">Element</a> of  <a href="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:29_1_1_2_2_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="sgraph1.html#T21" target="_self">Th21</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) as    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="sgraph1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E6:29_1_1_2_2_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>12</sub></font>]
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:29_1_1_2_2_1_2_2"><i><font color="Green">E28</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</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="Maroon">c<sub>12</sub></font></span>)</span><b> such that </b><br/><a NAME="E8:29_1_1_2_2_1_2_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>12</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>9</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E9:29_1_1_2_2_1_2_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P<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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> #)]
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:29"><i><font color="Green">E18</font></i></a>, <a class="txt" href="sgraph1.html#E1:29_1_1_2_2_1_2_2"><i><font color="Green">E28</font></i></a></i>;<br/><b>thus </b><a NAME="E10:29_1_1_2_2_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>8</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="setwiseo.html#K2">.}</a></span>]
 <b>by </b><i><a class="txt" href="sgraph1.html#E8:29_1_1_2_2_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sgraph1.html#E9:29_1_1_2_2_1_2_2"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:29_1_1_2_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">c<sub>7</sub></font> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S2">SETWISEO:sch 2</a>(<a class="txt" href="sgraph1.html#E3:29_1_1_2_2_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="sgraph1.html#E4:29_1_1_2_2_1_2"><i><font color="Green">E27</font></i></a>);<br/></i><b>hence </b><a NAME="E6:29_1_1_2_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>6</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font> #)]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:29_1_1_2"/><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:29_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E8:29_1_1_2"/>
the <a href="sgraph1.html#U1" target="_self">SEdges</a> of <font color="Maroon">c<sub>4</sub></font> is   <a href="finsub_1.html#NM1">Finite_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>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:29_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:29_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="sgraph1.html#E6:29_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="sgraph1.html#E7:29_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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