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<b>let </b><font color="Maroon" title="c1">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">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">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>] )</span><br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c1">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">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>]</span><br/>

<b>then consider </b><font color="Maroon" title="c2">v</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 <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c3">e</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> <font color="Maroon" title="c2">v</font></span>)</span><b> such that </b><br/><a NAME="E3:27_1"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">e</font> #)
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c1">g</font> as    <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="#E1:27_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="#E3:27_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="sgraph1.html#D7" target="_self" title="SGRAPH1:def.7">Def7</a></i>;<br/>
<a NAME="E5:27_1"/><i><font color="Green" title="E21">A6</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c4">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">G</font> )
 
<b>by </b><i><a class="ref" href="sgraph1.html#T23" target="_self" title="SGRAPH1:th.23">Th23</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>]</span><br/><div class="add"><a NAME="E2:27_1_1_1"/><b>then </b>
the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c4">G</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E5:27_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="sgraph1.html#T13" target="_self" title="SGRAPH1:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>]
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E5:27_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="#E1:27_1_1_1"><i><font color="Green" title="E22">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2"/>
not <font color="Maroon">F<sub>1</sub></font>() is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>]</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">X</font> = <font color="Maroon">F<sub>1</sub></font>() as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# $1,<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> $1</span>)</span></span>)</span> #)];<br/><a NAME="E3:27_1_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c5">X</font>]
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/><div><i><font color="Green" title="E23">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">B'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B'</font>] &amp; not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B'</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="b1">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><b>let </b><font color="Maroon" title="c7">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B'</font>] &amp; not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B'</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B'</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="c7">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/><b>assume </b><a NAME="E1:27_1_1_2_1"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B'</font>] &amp; not <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B'</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="c7">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><b>set </b><font color="Maroon" title="c8">g</font> =  <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">B'</font>,<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> <font color="Maroon" title="c6">B'</font></span>)</span></span>)</span> #);<br/><a NAME="E2:27_1_1_2_1"/>
<font color="Maroon" title="c6">B'</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="c5">X</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#T32" title="FINSUB_1:th.32">FINSUB_1:32</a></i>;<br/><a NAME="E3:27_1_1_2_1"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
 <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">B'</font>,<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> <font color="Maroon" title="c6">B'</font></span>)</span></span>)</span> #) <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">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">g</font> =  <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">B'</font>,<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> <font color="Maroon" title="c6">B'</font></span>)</span></span>)</span> #) as    <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="sgraph1.html#D7" target="_self" title="SGRAPH1:def.7">Def7</a></i>;<br/><a NAME="E5:27_1_1_2_1"/>
<font color="Maroon">P<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="Maroon" title="c9">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="c7">b</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="c9">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="c7">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> #)]
 <b>by </b><i><a class="txt" href="#E2:27"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E1:27_1_1_2_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="#E3:27_1_1_2_1"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E6:27_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B'</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="c7">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:27_1_1_2"/><i><font color="Green" title="E24">A12</font></i>: 
for <font color="Olive" title="b1">VV</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c5">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">VV</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="#E3:27_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="#E4:27_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>);<br/></i><div><i><font color="Green" title="E25">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">VV</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c5">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EEa</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Olive" title="b1">EEa</font></span>)</span><br/> for <font color="Olive" title="b2">EE</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> <font color="Olive" title="b1">EEa</font></span>)</span>  st <font color="Olive" title="b2">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Olive" title="b1">EEa</font>,<font color="Olive">b<sub>3</sub></font> #)]</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_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="E1:27_1_1_2_2_1"/>
(  <a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_2_2_1_1"/><i><font color="Green" title="E26">A14</font></i>: 
 <a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EEa</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Olive" title="b1">EEa</font></span>)</span><br/> for <font color="Olive" title="b2">EE</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> <font color="Olive" title="b1">EEa</font></span>)</span>  st <font color="Olive" title="b2">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Olive" title="b1">EEa</font>,<font color="Olive">b<sub>3</sub></font> #)]</span><br/><div class="add"><b>let </b><font color="Maroon" title="c7">EEa</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EE</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> <font color="Maroon" title="c6">VV</font></span>)</span>  st <font color="Maroon" title="c7">EEa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">EE</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Olive" title="b1">EE</font> #)]</span><br/><b>let </b><font color="Maroon" title="c8">EE</font> be   <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> <font color="Maroon" title="c6">VV</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">EEa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">EE</font> implies <font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c8">EE</font> #)] )</span><br/><b>assume </b><a NAME="E2:27_1_1_2_2_1_1"/>
<font color="Maroon" title="c7">EEa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">EE</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c8">EE</font> #)]</span><br/><a NAME="E3:27_1_1_2_2_1_1"/>
<font color="Maroon" title="c8">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_2_2_1_1"><i><font color="Green" title="E26">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E4:27_1_1_2_2_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c8">EE</font> #)]
 <b>by </b><i><a class="txt" href="#E5:27_1_1_2"><i><font color="Green" title="E24">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2_2_1_2"/>
 <a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EE</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Olive" title="b1">EE</font></span>)</span><br/> for <font color="Olive" title="b2">EE'</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> <font color="Olive" title="b1">EE</font></span>)</span>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">EE'</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Olive" title="b1">EE</font>,<font color="Olive">b<sub>3</sub></font> #)]</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">TT</font> =  <a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c7">TT</font>] <b>means </b>for <font color="Olive" title="b1">EE</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> <font color="Maroon" title="c6">VV</font></span>)</span>  st <font color="Olive" title="b1">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Olive" title="b1">EE</font> #)];<br/><a NAME="E3:27_1_1_2_2_1_2"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c7">TT</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">EE</font> be   <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> <font color="Maroon" title="c6">VV</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c7">TT</font> implies <font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c8">EE</font> #)] )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c8">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c7">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c8">EE</font> #)]</span><br/>

<a NAME="E2:27_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">EE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:27_1_1_2_2_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c8">EE</font> #)]
 <b>by </b><i><a class="txt" href="#E5:27_1_1_2"><i><font color="Green" title="E24">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><div><i><font color="Green" title="E27">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">ee</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c7">TT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">vv</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">TT</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">ee</font>] &amp; not <font color="Olive" title="b1">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">ee</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">ee</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">vv</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><b>let </b><font color="Maroon" title="c9">vv</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">TT</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">ee</font>] &amp; not <font color="Maroon" title="c9">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">ee</font> implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">ee</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">vv</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/><b>assume </b><a NAME="E1:27_1_1_2_2_1_2_2"/><i><font color="Green" title="E28">A17</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">ee</font>] &amp; not <font color="Maroon" title="c9">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">ee</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">ee</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">vv</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><b>reconsider </b><font color="Maroon" title="c10">ee'</font> = <font color="Maroon" title="c8">ee</font> as   <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> <font color="Maroon" title="c6">VV</font></span>)</span> <b>by </b><i><a class="ref" href="finsub_1.html#T32" title="FINSUB_1:th.32">FINSUB_1:32</a></i>;<br/><a NAME="E3:27_1_1_2_2_1_2_2"/><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c6">VV</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">F<sub>1</sub></font>()
 <b>by </b><i><a class="ref" href="finsub_1.html#T32" title="FINSUB_1:th.32">FINSUB_1:32</a></i>;<br/><b>set </b><font color="Maroon" title="c11">g</font> =  <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c10">ee'</font> #);<br/><a NAME="E4:27_1_1_2_2_1_2_2"/>
 <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c10">ee'</font> #) is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="sgraph1.html#K3" title="SGRAPH1:func.3">SIMPLEGRAPHS</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="#E3:27_1_1_2_2_1_2_2"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="sgraph1.html#T21" target="_self" title="SGRAPH1:th.21">Th21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">g</font> =  <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Maroon" title="c10">ee'</font> #) as    <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="sgraph1.html#D7" target="_self" title="SGRAPH1:def.7">Def7</a></i>;<br/><a NAME="E6:27_1_1_2_2_1_2_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c12">g</font>]
 <b>by </b><i><a class="txt" href="#E1:27_1_1_2_2_1_2_2"><i><font color="Green" title="E28">A17</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">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="Maroon" title="c12">g</font></span>)</span><b> such that </b><br/><a NAME="E8:27_1_1_2_2_1_2_2"/><i><font color="Green" title="E30">A19</font></i>: 
<font color="Maroon" title="c13">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="Maroon" title="c12">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="c9">vv</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E9:27_1_1_2_2_1_2_2"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon">P<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="Maroon" title="c12">g</font>,<font color="Maroon" title="c13">sege</font> #)]
 <b>by </b><i><a class="txt" href="#E3:27"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="#E1:27_1_1_2_2_1_2_2"><i><font color="Green" title="E28">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E10:27_1_1_2_2_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">ee</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">vv</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 <b>by </b><i><a class="txt" href="#E8:27_1_1_2_2_1_2_2"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="#E9:27_1_1_2_2_1_2_2"><i><font color="Green" title="E31">A20</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:27_1_1_2_2_1_2"/>
for <font color="Olive" title="b1">EE</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c7">TT</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">EE</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="#E3:27_1_1_2_2_1_2"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="#E4:27_1_1_2_2_1_2"><i><font color="Green" title="E27">A16</font></i></a>);<br/></i><b>hence </b><a NAME="E6:27_1_1_2_2_1_2"/>
for <font color="Olive" title="b1">EE</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" title="SGRAPH1:func.2">TWOELEMENTSETS</a> <font color="Maroon" title="c6">VV</font></span>)</span><br/> for <font color="Olive" title="b2">EE'</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> <font color="Maroon" title="c6">VV</font></span>)</span>  st <font color="Olive" title="b2">EE'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">EE</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="sgraph1.html#G1" title="SGRAPH1:aggr.1">SimpleGraphStruct</a>(# <font color="Maroon" title="c6">VV</font>,<font color="Olive" title="b2">EE'</font> #)]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:27_1_1_2"/><i><font color="Green" title="E26">A21</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="#E3:27_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E8:27_1_1_2"/>
the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c4">G</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_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="c4">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:27_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">g</font>]
 <b>by </b><i><a class="txt" href="#E6:27_1_1_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="#E7:27_1_1_2"><i><font color="Green" title="E26">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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