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

<b>let </b><font color="Maroon" title="c1">X</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"> 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> <br/> for <font color="Olive" title="b3">vg</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">vg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="Olive" title="b3">vg</font> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b3">vg</font> holds <br/>for <font color="Olive" title="b4">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">vg</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">w</font> holds <br/> ex <font color="Olive" title="b5">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b5">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> &amp; <font color="Olive" title="b5">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="Olive" title="b2">v</font>,<font color="Olive" title="b4">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><b>let </b><font color="Maroon" title="c2">g</font> be    <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon" title="c1">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">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">vg</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">vg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">vg</font> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Maroon" title="c2">g</font>,<font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">vg</font> holds <br/>for <font color="Olive" title="b3">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">vg</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w</font> holds <br/> ex <font color="Olive" title="b4">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">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="c2">g</font> &amp; <font color="Olive" title="b4">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="Olive" title="b1">v</font>,<font color="Olive" title="b3">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><b>let </b><font color="Maroon" title="c3">v</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"> for <font color="Olive" title="b1">vg</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">vg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">vg</font> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">vg</font> holds <br/>for <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">vg</font>  st <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">w</font> holds <br/> ex <font color="Olive" title="b3">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">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="c2">g</font> &amp; <font color="Olive" title="b3">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="c3">v</font>,<font color="Olive" title="b2">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><b>let </b><font color="Maroon" title="c4">vg</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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="c4">vg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">vg</font> implies for <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">vg</font>  st <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">w</font> holds <br/> ex <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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="Maroon" title="c2">g</font> &amp; <font color="Olive" title="b2">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="c3">v</font>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )</span><br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c4">vg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font> &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">vg</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">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">vg</font>  st <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">w</font> holds <br/> ex <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <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="Maroon" title="c2">g</font> &amp; <font color="Olive" title="b2">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="c3">v</font>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<a NAME="E2:39"/><b>then </b>
<font color="Maroon" title="c4">vg</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="sgraph1.html#T27" target="_self" title="SGRAPH1:th.27">Th27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c1">X</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>  <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">vg</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font> implies  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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="c2">g</font> &amp; <font color="Olive" title="b1">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="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )</span><br/>

<b>assume </b><a NAME="E4:39"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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="c2">g</font> &amp; <font color="Olive" title="b1">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="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<b>take </b>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c2">g</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>
<div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
not <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c2">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c7">ww</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c7">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">vv</font> where <font color="Olive" title="b1">vv</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font> : ( <font color="Olive" title="b1">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Olive" title="b1">vv</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c2">g</font> ) </span>}</span>  &amp;  <a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">ww</font> )
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="sgraph1.html#T39" target="_self" title="SGRAPH1:th.39">Th39</a></i>;<br/><div><i><font color="Green" title="E27">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">e</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="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">ww</font> implies <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font> )</span><br/><b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">ww</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font></span><br/><b>then consider </b><font color="Maroon" title="c9">vv</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font><b> such that </b><br/><a NAME="E3:39_1_1_1"/><i><font color="Green" title="E28">A6</font></i>: 
( <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">vv</font> &amp; <font color="Maroon" title="c9">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c9">vv</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c2">g</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><b>thus </b><a NAME="E4:39_1_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font>
 <b>by </b><i><a class="txt" href="#E3:39_1_1_1"><i><font color="Green" title="E28">A6</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="E5:39_1_1"/><i><font color="Green" title="E28">A7</font></i>: 
( not <font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">ww</font> &amp; not <font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">ww</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">ww</font></span>
<div class="add"><b>assume </b><a NAME="E1:39_1_1_2_1"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">ww</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c8">vv</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font><b> such that </b><br/><a NAME="E3:39_1_1_2_1"/><i><font color="Green" title="E29">A8</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vv</font> &amp; <font color="Maroon" title="c8">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c8">vv</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c2">g</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><a NAME="E4:39_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c2">g</font>
 <b>by </b><i><a class="txt" href="#E3:39_1_1_2_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">z</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" title="c4">vg</font><b> such that </b><br/><a NAME="E6:39_1_1_2_1"/><i><font color="Green" title="E30">A9</font></i>: 
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="sgraph1.html#T9" target="_self" title="SGRAPH1:th.9">Th9</a></i>;<br/><b>thus </b><a NAME="E7:39_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:39_1_1_2_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</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>

<b>assume </b><a NAME="E2:39_1_1_2"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">ww</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c8">vv</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X</font><b> such that </b><br/><a NAME="E4:39_1_1_2"/><i><font color="Green" title="E29">A10</font></i>: 
( <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vv</font> &amp; <font color="Maroon" title="c8">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c8">vv</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c2">g</font> )
 <b>by </b><i><a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:39_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:39_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E4:39_1_1_2"><i><font color="Green" title="E29">A10</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><b>reconsider </b><font color="Maroon" title="c8">wwv</font> = <font color="Maroon" title="c7">ww</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="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E7:39_1_1"/><i><font color="Green" title="E29">A11</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">wwv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">ww</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:39_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/><a NAME="E8:39_1_1"/>
( <font color="Maroon" title="c8">wwv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">vg</font> &amp; <font color="Maroon" title="c8">wwv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">vg</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_3"/><i><font color="Green" title="E30">A12</font></i>: 
<font color="Maroon" title="c7">ww</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">vg</font>
 
<b>by </b><i><a class="txt" href="#E4:39_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E2:39_1_1_3"/>
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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">vg</font>
 
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:39_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">vg</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_1_3"/>
<font color="Maroon" title="c8">wwv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">vg</font>
 <b>by </b><i><a class="txt" href="#E1:39_1_1_3"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">wwv</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">vg</font></span><br/>

<a NAME="E5:39_1_1_3"/><i><font color="Green" title="E31">A13</font></i>: 
not <font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E4:39"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E6:39_1_1_3"/>
<font color="Maroon" title="c8">wwv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">vg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>hence </b><a NAME="E7:39_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E5:39_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="#E5:39_1_1_3"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="E9:39_1_1"/><b>then </b>
<font color="Maroon" title="c8">wwv</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">vg</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E10:39_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E7:39_1_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</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><b>hence </b><a NAME="E2:39_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c2">g</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>

<b>thus </b><a NAME="E6:39"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</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>
