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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="chord.html#V6" target="_self">chordal</a> implies  ex <font color="Olive">S</font> being   <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">a<sub>1</sub></font> st <font color="Olive">S</font> is <a href="chord.html#V7" target="_self">perfect</a> );<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:135_1_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">Gk</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>  st <font color="Olive">Gk</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Gk</font>]
 ;<br/><b>let </b><font color="Maroon">Gk1</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>assume </b><a NAME="E2:135_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Gk1</font> <a href="glib_000.html#K16">.order()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><b>thus </b><a NAME="E3:135_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Gk1</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="135_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:135_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">Gk1</font> is <a href="chord.html#V6" target="_self">chordal</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:135_1_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:135_1_1_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:135_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">Gk1</font> is <a href="glib_000.html#V4">trivial</a>
 <b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="glib_000.html#T29">GLIB_000:29</a></i>;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gk1</font>;<br/><a NAME="E4:135_1_1_1_1_1_1"/>
 ex <font color="Olive">S</font> being   <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">Gk1</font> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Olive">S</font> is <a href="chord.html#V7" target="_self">perfect</a> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:135_1_1_1_1_1_1"/>
 ex <font color="Olive">S</font> being   <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">Gk1</font> st <font color="Olive">S</font> is <a href="chord.html#V7" target="_self">perfect</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:135_1_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">Gk1</font> as  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="chord.html#V6" target="_self">chordal</a> <a href="glib_000.html#NM2">_Graph</a> <b>by </b><i><a class="ref" href="chord.html#T24" target="_self">Th24</a>, <a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="glib_000.html#T29">GLIB_000:29</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:135_1_1_1_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">H</font> being   <a href="glib_000.html#NM6">removeVertex</a> of <font color="Maroon">G</font>,<font color="Maroon">x</font>;<br/><a NAME="E6:135_1_1_1_1_1_2"/><i><font color="Green">E67</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="glib_000.html#T23">GLIB_000:23</a></i>;<br/><a NAME="E7:135_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><font color="Maroon">H</font> <a href="glib_000.html#K16">.order()</a>  = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="chord.html#T23" target="_self">Th23</a>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>
;<br/><a NAME="E9:135_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">H</font> <a href="glib_000.html#K16">.order()</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><b>consider </b><font color="Maroon">T</font> being    <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">H</font><b> such that </b><br/><a NAME="E11:135_1_1_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">T</font> is <a href="chord.html#V7" target="_self">perfect</a>
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a>, </i>;<br/><b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">T</font>;<br/><a NAME="E12:135_1_1_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:135_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E14:135_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E15:135_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</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="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/><a NAME="E16:135_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">T</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">T</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/><a NAME="E19:135_1_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E20:135_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="subset_1.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E21:135_1_1_1_1_1_2"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_3.html#T102">FINSEQ_3:102</a></i>;<br/><a NAME="E22:135_1_1_1_1_1_2"/>
<font color="Maroon">T</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/><a NAME="E23:135_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">S</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T98">FINSEQ_3:98</a></i>;<br/><b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as    <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">Gk1</font> <b>by </b><i>, </i>;<br/><b>take </b>
<font color="Maroon">S</font>
;<br/><b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D13" target="_self">CHORD:def 13</a><br/><b>assume </b><a NAME="E25:135_1_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 ;<br/><b>let </b><font color="Maroon">Gf</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">Gk1</font>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">Gf</font>;<br/><b>assume </b><a NAME="E26:135_1_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E27:135_1_1_1_1_1_2"/><i><font color="Green">E76</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1_1_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:135_1_1_1_1_1_2_3"/>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1_1_1_2_3_1"><b>suppose </b></a><a NAME="E1:135_1_1_1_1_1_2_3_1"/><i><font color="Green">E80</font></i>: 
1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:135_1_1_1_1_1_2_3_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n</font> = 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#T7">GRAPH_2:7</a></i>
<br/>.= 
 <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
<b>by </b><i/>
;<br/><a NAME="E4:135_1_1_1_1_1_2_3_1"/><b>then </b>
<font color="Maroon">Gf</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T97">GLIB_000:97</a></i>;<br/><b>hence </b><a NAME="E5:135_1_1_1_1_1_2_3_1"/>
<font color="Maroon">v</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="135_1_1_1_1_1_2_3_2"><b>suppose </b></a><a NAME="E1:135_1_1_1_1_1_2_3_2"/><i><font color="Green">E82</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:135_1_1_1_1_1_2_3_2"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 as  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E4:135_1_1_1_1_1_2_3_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E5:135_1_1_1_1_1_2_3_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/><a NAME="E6:135_1_1_1_1_1_2_3_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E7:135_1_1_1_1_1_2_3_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:135_1_1_1_1_1_2_3_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E9:135_1_1_1_1_1_2_3_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">T</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="chord.html#K4" target="_self">.followSet</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:135_1_1_1_1_1_2_3_2"/>
not <font color="Maroon">T</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, </i>;<br/><a NAME="E11:135_1_1_1_1_1_2_3_2"/><b>then </b>
<font color="Maroon">Gf</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">H</font>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="chord.html#K4" target="_self">.followSet</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="chord.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E12:135_1_1_1_1_1_2_3_2"/>
<font color="Maroon">v</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 <b>by </b><i>, , , <a class="ref" href="chord.html#T25" target="_self">Th25</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:135_1_1"/>
for <font color="Olive">G</font> being <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">G</font>]
 
<b>from </b><i><a class="ref" href="chord.html#S1" target="_self">CHORD:sch 1</a>(<a class="ref" href="chord.html#T21" target="_self">Th21</a>);<br/></i>
<b>then consider </b><font color="Maroon">S</font> being    <a href="chord.html#M3" target="_self">VertexScheme</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:135_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">S</font> is <a href="chord.html#V7" target="_self">perfect</a>
 ;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E5:135_1_1"/>
<font color="Maroon">S</font> is <a href="chord.html#V7" target="_self">perfect</a>
 <b>by </b><i><a class="ref" href="chord.html#T22" target="_self">Th22</a></i>;<br/>


</div>
