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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<a NAME="E2:93"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> : verum </span>}</span> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:93_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font><b> such that </b><br/><a NAME="E3:93_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><b>thus </b><a NAME="E4:93_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:93"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="chord.html#T17" target="_self">Th17</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">p</font> being   <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">p</font> );<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:93_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font><b> such that </b><br/><a NAME="E3:93_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E4:93_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>hence </b><a NAME="E5:93_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:93"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E9:93"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Min</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:93"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">N</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">Min</font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">N</font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2">GRAPH_5:sch 2</a>();<br/></i>
<b>consider </b><font color="Maroon">M</font> being    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font><b> such that </b><br/><a NAME="E13:93"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Min</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Min</font> )
 <b>by </b><i/>;<br/>
<a NAME="E14:93"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:93"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_3_1"/>
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">M</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="93_3_1_1"><b>suppose </b></a><a NAME="E1:93_3_1_1"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:93_3_1_1"/>
<font color="Maroon">M</font> is <a href="chord.html#V4" target="_self">minimal</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_3_1_2"><b>suppose </b></a><a NAME="E1:93_3_1_2"/>
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_3_1_2_1"/>
not <font color="Maroon">M</font> is <a href="chord.html#V4" target="_self">minimal</a>
 ;<br/><b>then consider </b><font color="Maroon">T</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:93_3_1_2_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">T</font> is    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="chord.html#E10:93_3_1_2_1"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E4:93_3_1_2_1"/>
<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">T2</font> = <font color="Maroon">T</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/><b>consider </b><font color="Maroon">Tp</font> being    <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font><b> such that </b><br/><a NAME="E7:93_3_1_2_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">Tp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">Tp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">T2</font> )
 <b>by </b><i/>;<br/><a NAME="E8:93_3_1_2_1"/>
<font color="Maroon">Tp</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E9:93_3_1_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">Tp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Tp</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:93_3_1_2_1"/><i><font color="Green">E74</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">M</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">T</font>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E11:93_3_1_2_1"/><i><font color="Green">E75</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">T</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E12:93_3_1_2_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/><b>hence </b><a NAME="E13:93_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="chord.html#T18" target="_self">Th18</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:93_3_1_2"/>
<font color="Maroon">M</font> is <a href="chord.html#V4" target="_self">minimal</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:93"/>
 ex <font color="Olive">S</font> being   <a href="chord.html#M2" target="_self">VertexSeparator</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> st <font color="Olive">S</font> is <a href="chord.html#V4" target="_self">minimal</a>
 ;<br/>


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