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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">OTHER</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">OTHER</font>;<br/>
<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">OTHER</font> <a href="sgraph1.html#R2" title="SGRAPH1:pred.2">is_SetOfSimpleGraphs_of</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p2">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span> #)]
 
<b>by </b><i><a class="ref" href="sgraph1.html#D9" target="_self" title="SGRAPH1:def.9">Def9</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green" title="E23">A3</font></i>: 
for <font color="Olive" title="b1">g</font> being   <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K3" title="SGRAPH1:func.3">SIMPLEGRAPHS</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">g</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p2">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="c3">g</font> be    <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K3" title="SGRAPH1:func.3">SIMPLEGRAPHS</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">OTHER</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">SVg</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font>;<br/>
<b>set </b><font color="Maroon" title="c6">SEg</font> = the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c3">g</font>;<br/>
<a NAME="E2:35_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> &amp; the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c3">g</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sgraph1.html#T27" target="_self" title="SGRAPH1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E3:35_1"/>
 <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p2">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">OTHER</font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="sgraph1.html#E1:35_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="sgraph1.html#D9" target="_self" title="SGRAPH1:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:35"/><i><font color="Green" title="E24">A5</font></i>: 
for <font color="Olive" title="b1">g</font> being   <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">g</font>] &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">g</font> &amp; not <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Olive" title="b1">g</font> holds <br/> ex <font color="Olive" title="b3">sege</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">g</font></span>)</span> st <br/>( <font color="Olive" title="b3">sege</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Olive" title="b1">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">g</font>,<font color="Olive" title="b3">sege</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" title="c3">g</font> be    <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:35_2"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">OTHER</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> &amp; not <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c3">g</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">SVg</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font>;<br/>
<b>set </b><font color="Maroon" title="c6">SEg</font> = the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c3">g</font>;<br/>
<a NAME="E2:35_2"/><i><font color="Green" title="E26">A7</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> &amp; the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c3">g</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sgraph1.html#T27" target="_self" title="SGRAPH1:th.27">Th27</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:35_2"/><i><font color="Green" title="E27">A8</font></i>: 
( <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c7">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v2</font> &amp; <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c8">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:35_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="sgraph1.html#T10" target="_self" title="SGRAPH1:th.10">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">v1v2</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font></span>)</span><b> such that </b><br/><a NAME="E6:35_2"/><i><font color="Green" title="E28">A9</font></i>: 
( <font color="Maroon" title="c9">v1v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c3">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c8">v2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c9">v1v2</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">OTHER</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:35"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="sgraph1.html#E1:35_2"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="sgraph1.html#E2:35_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="sgraph1.html#E4:35_2"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="sgraph1.html#D9" target="_self" title="SGRAPH1:def.9">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">v1v2</font>
;<br/>

<b>thus </b><a NAME="E7:35_2"/>
( <font color="Maroon" title="c9">v1v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c3">g</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c9">v1v2</font> #) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">OTHER</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E4:35_2"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="sgraph1.html#E6:35_2"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:35"/><i><font color="Green" title="E25">A10</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K3" title="SGRAPH1:func.3">SIMPLEGRAPHS</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>]
 
<b>from </b><i><a class="ref" href="sgraph1.html#S1" target="_self" title="SGRAPH1:sch.1">SGRAPH1:sch 1</a>(<a class="txt" href="sgraph1.html#E2:35"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="sgraph1.html#E3:35"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="sgraph1.html#E4:35"><i><font color="Green" title="E24">A5</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:35"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sgraph1.html#K3" title="SGRAPH1:func.3">SIMPLEGRAPHS</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>hence </b><a NAME="E7:35"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">OTHER</font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E5:35"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/>


</div>
