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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be    <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font>;<br/>



<a NAME="E1:134"/><i><font color="Green" title="E100">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="chord.html#D12" target="_self" title="CHORD:def.12">Def12</a></i>;<br/>
<a NAME="E2:134"/><i><font color="Green" title="E101">A2</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="chord.html#D12" target="_self" title="CHORD:def.12">Def12</a></i>;<br/>
<div><i><font color="Green" title="E102">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:134_1"/><i><font color="Green" title="E103">A4</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 ;<br/><b>let </b><font color="Maroon" title="c6">va</font>, <font color="Maroon" title="c7">vb</font>, <font color="Maroon" title="c8">vc</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E2:134_1"/><i><font color="Green" title="E104">A5</font></i>: 
( <font color="Maroon" title="c6">va</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c7">vb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c8">vc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">vb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">vc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c6">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">vb</font> &amp; <font color="Maroon" title="c6">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">vc</font> )
 ;<br/><a NAME="E3:134_1"/><i><font color="Green" title="E105">A6</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="txt" href="chord.html#E2:134"><i><font color="Green" title="E101">A2</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>assume </b><a NAME="E4:134_1"/><i><font color="Green" title="E106">A7</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a>
 ;<br/><a NAME="E5:134_1"/><i><font color="Green" title="E107">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">va</font> &amp; <font color="Maroon" title="c6">va</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font> )
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:134_1"/><b>then </b><i><font color="Green" title="E108">A9</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:134_1"/><i><font color="Green" title="E109">A10</font></i>: 
not <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c6">va</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="chord.html#E5:134_1"><i><font color="Green" title="E107">A8</font></i></a>, <a class="txt" href="chord.html#E6:134_1"><i><font color="Green" title="E108">A9</font></i></a>, <a class="ref" href="chord.html#T107" target="_self" title="CHORD:th.107">Th107</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">Gf</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c6">va</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">fs</font> = <font color="Maroon" title="c6">va</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font>;<br/><a NAME="E9:134_1"/><i><font color="Green" title="E110">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">va</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:134_1"><i><font color="Green" title="E107">A8</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>;<br/><a NAME="E10:134_1"/><i><font color="Green" title="E111">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E9:134_1"><i><font color="Green" title="E110">A11</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">jb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:134_1"/><i><font color="Green" title="E112">A13</font></i>: 
<font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">jb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vb</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">jc</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:134_1"/><i><font color="Green" title="E113">A14</font></i>: 
<font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">jc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vc</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E15:134_1"/>
<font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="txt" href="chord.html#E5:134_1"><i><font color="Green" title="E107">A8</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:134_1"/><b>then </b><i><font color="Green" title="E114">A15</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T41" title="XREAL_1:th.41">XREAL_1:41</a>, <a class="txt" href="chord.html#E10:134_1"><i><font color="Green" title="E111">A12</font></i></a></i>;<br/><a NAME="E17:134_1"/><b>then </b><i><font color="Green" title="E115">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0</span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E5:134_1"><i><font color="Green" title="E107">A8</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>;<br/><div><i><font color="Green" title="E116">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1"/>
<font color="Maroon" title="c11">jb</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ;<br/><a NAME="E2:134_1_1"/><b>then </b>
<font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">jb</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:134_1"><i><font color="Green" title="E110">A11</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E3:134_1_1"/><b>then </b>
<font color="Maroon" title="c7">vb</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E12:134_1"><i><font color="Green" title="E112">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:134_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><a NAME="E19:134_1"/><i><font color="Green" title="E117">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">jb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">jb</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E18:134_1"><i><font color="Green" title="E116">A17</font></i></a>, <a class="txt" href="chord.html#E5:134_1"><i><font color="Green" title="E107">A8</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>;<br/><div><i><font color="Green" title="E118">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_2"/>
<font color="Maroon" title="c12">jc</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ;<br/><a NAME="E2:134_1_2"/><b>then </b>
<font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">jc</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="txt" href="chord.html#E9:134_1"><i><font color="Green" title="E110">A11</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E3:134_1_2"/><b>then </b>
<font color="Maroon" title="c8">vc</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E14:134_1"><i><font color="Green" title="E113">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E4:134_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><a NAME="E21:134_1"/><i><font color="Green" title="E119">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">jc</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">jc</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E20:134_1"><i><font color="Green" title="E118">A19</font></i></a>, <a class="txt" href="chord.html#E5:134_1"><i><font color="Green" title="E107">A8</font></i></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></i>;<br/><a NAME="E22:134_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E16:134_1"><i><font color="Green" title="E114">A15</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E23:134_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:134_1"/><b>then </b><i><font color="Green" title="E120">A21</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="txt" href="chord.html#E17:134_1"><i><font color="Green" title="E115">A16</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E25:134_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">jb</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E26:134_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">jb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E27:134_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">jb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">jb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span> )
 <b>by </b><i><a class="txt" href="chord.html#E18:134_1"><i><font color="Green" title="E116">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E28:134_1"/><b>then </b>
<font color="Maroon" title="c11">jb</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E29:134_1"/><b>then </b><i><font color="Green" title="E121">A22</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="txt" href="chord.html#E19:134_1"><i><font color="Green" title="E117">A18</font></i></a>, <a class="txt" href="chord.html#E12:134_1"><i><font color="Green" title="E112">A13</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E30:134_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">jc</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E31:134_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">jc</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E32:134_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">jc</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">jc</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span> )
 <b>by </b><i><a class="txt" href="chord.html#E20:134_1"><i><font color="Green" title="E118">A19</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E33:134_1"/><b>then </b>
<font color="Maroon" title="c12">jc</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">va</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:134_1"/><b>then </b><i><font color="Green" title="E122">A23</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c6">va</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="txt" href="chord.html#E21:134_1"><i><font color="Green" title="E119">A20</font></i></a>, <a class="txt" href="chord.html#E14:134_1"><i><font color="Green" title="E113">A14</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">ag</font> = <font color="Maroon" title="c3">a</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">Gf</font> <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a>, <a class="txt" href="chord.html#E24:134_1"><i><font color="Green" title="E120">A21</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">bg</font> = <font color="Maroon" title="c4">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">Gf</font> <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a>, <a class="txt" href="chord.html#E29:134_1"><i><font color="Green" title="E121">A22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">cg</font> = <font color="Maroon" title="c5">c</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c9">Gf</font> <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a>, <a class="txt" href="chord.html#E34:134_1"><i><font color="Green" title="E122">A23</font></i></a></i>;<br/><a NAME="E38:134_1"/><i><font color="Green" title="E123">A24</font></i>: 
<font color="Maroon" title="c13">ag</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a>
 <b>by </b><i><a class="txt" href="chord.html#E4:134_1"><i><font color="Green" title="E106">A7</font></i></a>, <a class="txt" href="chord.html#E6:134_1"><i><font color="Green" title="E108">A9</font></i></a>, <a class="txt" href="chord.html#E5:134_1"><i><font color="Green" title="E107">A8</font></i></a>, <a class="txt" href="chord.html#E2:134_1"><i><font color="Green" title="E104">A5</font></i></a>, <a class="ref" href="chord.html#D13" target="_self" title="CHORD:def.13">Def13</a></i>;<br/><a NAME="E39:134_1"/><i><font color="Green" title="E124">A25</font></i>: 
( <font color="Maroon" title="c13">ag</font>,<font color="Maroon" title="c14">bg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c13">ag</font>,<font color="Maroon" title="c15">cg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c14">bg</font>,<font color="Maroon" title="c15">cg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E7:134_1"><i><font color="Green" title="E109">A10</font></i></a>, <a class="txt" href="chord.html#E1:134_1"><i><font color="Green" title="E103">A4</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></i>;<br/><a NAME="E40:134_1"/><b>then </b>
( <font color="Maroon" title="c14">bg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gf</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">ag</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c15">cg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Gf</font> <a href="chord.html#K2" title="CHORD:func.2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">ag</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="chord.html#E3:134_1"><i><font color="Green" title="E105">A6</font></i></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></i>;<br/><a NAME="E41:134_1"/><b>then </b>
 ex <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="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c14">bg</font>,<font color="Maroon" title="c15">cg</font>,<font color="Maroon" title="c9">Gf</font>
 <b>by </b><i><a class="txt" href="chord.html#E38:134_1"><i><font color="Green" title="E123">A24</font></i></a>, <a class="txt" href="chord.html#E1:134_1"><i><font color="Green" title="E103">A4</font></i></a>, <a class="ref" href="chord.html#T68" target="_self" title="CHORD:th.68">Th68</a></i>;<br/><b>hence </b><a NAME="E42:134_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E39:134_1"><i><font color="Green" title="E124">A25</font></i></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_2"/>
not <font color="Maroon" title="c2">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a>
 ;<br/><a NAME="E2:134_2"/><b>then </b>
 ex <font color="Olive" title="b1">n</font> being non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font> &amp;  ex <font color="Olive" title="b2">Gf</font> being  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Olive" title="b1">n</font></span>)</span> ex <font color="Olive" title="b3">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">Gf</font> st <br/>( <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; not <font color="Olive" title="b3">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> ) )
 <b>by </b><i><a class="ref" href="chord.html#D13" target="_self" title="CHORD:def.13">Def13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">n</font> being  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>, <font color="Maroon" title="c4">Gf</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">n</font></span>)</span>, <font color="Maroon" title="c5">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">Gf</font><b> such that </b><br/><a NAME="E4:134_2"/><i><font color="Green" title="E103">A26</font></i>: 
( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> &amp; not <font color="Maroon" title="c5">v</font> is <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> )
 ;<br/><a NAME="E5:134_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">V</font> )
 <b>by </b><i><a class="txt" href="chord.html#E4:134_2"><i><font color="Green" title="E103">A26</font></i></a>, <a class="ref" href="chord.html#T1" target="_self" title="CHORD:th.1">Th1</a></i>;<br/><a NAME="E6:134_2"/><b>then </b><i><font color="Green" title="E104">A27</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">Gf</font><b> such that </b><br/><a NAME="E8:134_2"/><i><font color="Green" title="E105">A28</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> )
 <b>and </b><br/><a NAME="E9:134_2"/><i><font color="Green" title="E106">A29</font></i>: 
( <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">a</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c7">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E4:134_2"><i><font color="Green" title="E103">A26</font></i></a>, <a class="ref" href="chord.html#T69" target="_self" title="CHORD:th.69">Th69</a></i>;<br/><a NAME="E10:134_2"/><i><font color="Green" title="E107">A30</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">n</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E4:134_2"><i><font color="Green" title="E103">A26</font></i></a>, <a class="ref" href="chord.html#T107" target="_self" title="CHORD:th.107">Th107</a></i>;<br/><a NAME="E11:134_2"/><b>then </b><i><font color="Green" title="E108">A31</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">Gf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> <a href="chord.html#K4" title="CHORD:func.4">.followSet</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">vg</font> = <font color="Maroon" title="c5">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E13:134_2"/><i><font color="Green" title="E109">A32</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">ag</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c10">bg</font> = <font color="Maroon" title="c7">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E15:134_2"/><i><font color="Green" title="E110">A33</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E16:134_2"/><i><font color="Green" title="E111">A34</font></i>: 
( <font color="Maroon" title="c8">vg</font>,<font color="Maroon" title="c9">ag</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c8">vg</font>,<font color="Maroon" title="c10">bg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; not <font color="Maroon" title="c9">ag</font>,<font color="Maroon" title="c10">bg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E10:134_2"><i><font color="Green" title="E107">A30</font></i></a>, <a class="txt" href="chord.html#E9:134_2"><i><font color="Green" title="E106">A29</font></i></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></i>;<br/><a NAME="E17:134_2"/><i><font color="Green" title="E112">A35</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="txt" href="chord.html#E1:134"><i><font color="Green" title="E100">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">va</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:134_2"/><i><font color="Green" title="E113">A36</font></i>: 
( <font color="Maroon" title="c11">va</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="txt" href="chord.html#E13:134_2"><i><font color="Green" title="E109">A32</font></i></a>, <a class="txt" href="chord.html#E1:134"><i><font color="Green" title="E100">A1</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><a NAME="E20:134_2"/><i><font color="Green" title="E114">A37</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="txt" href="chord.html#E1:134"><i><font color="Green" title="E100">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">vb</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E22:134_2"/><i><font color="Green" title="E115">A38</font></i>: 
( <font color="Maroon" title="c12">vb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">vb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="txt" href="chord.html#E15:134_2"><i><font color="Green" title="E110">A33</font></i></a>, <a class="txt" href="chord.html#E1:134"><i><font color="Green" title="E100">A1</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><div><i><font color="Green" title="E116">A39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_2_1"/>
<font color="Maroon" title="c11">va</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ;<br/><a NAME="E2:134_2_1"/><b>then </b><i><font color="Green" title="E117">A40</font></i>: 
<font color="Maroon" title="c11">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:134_2"><i><font color="Green" title="E105">A28</font></i></a>, <a class="txt" href="chord.html#E4:134_2"><i><font color="Green" title="E103">A26</font></i></a>, <a class="txt" href="chord.html#E19:134_2"><i><font color="Green" title="E113">A36</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:134_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T16" target="_self" title="CHORD:th.16">Th16</a>, <a class="txt" href="chord.html#E17:134_2"><i><font color="Green" title="E112">A35</font></i></a>, <a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="txt" href="chord.html#E6:134_2"><i><font color="Green" title="E104">A27</font></i></a>, <a class="txt" href="chord.html#E2:134"><i><font color="Green" title="E101">A2</font></i></a></i>;<br/><a NAME="E4:134_2_1"/><b>then </b>
<font color="Maroon" title="c11">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:134_2_1"><i><font color="Green" title="E117">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:134_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E19:134_2"><i><font color="Green" title="E113">A36</font></i></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E117">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_2_2"/>
<font color="Maroon" title="c12">vb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ;<br/><a NAME="E2:134_2_2"/><b>then </b><i><font color="Green" title="E118">A42</font></i>: 
<font color="Maroon" title="c12">vb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E8:134_2"><i><font color="Green" title="E105">A28</font></i></a>, <a class="txt" href="chord.html#E4:134_2"><i><font color="Green" title="E103">A26</font></i></a>, <a class="txt" href="chord.html#E22:134_2"><i><font color="Green" title="E115">A38</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:134_2_2"/>
<font color="Maroon" title="c7">b</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">V</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="chord.html#T16" target="_self" title="CHORD:th.16">Th16</a>, <a class="txt" href="chord.html#E20:134_2"><i><font color="Green" title="E114">A37</font></i></a>, <a class="txt" href="chord.html#E11:134_2"><i><font color="Green" title="E108">A31</font></i></a>, <a class="txt" href="chord.html#E6:134_2"><i><font color="Green" title="E104">A27</font></i></a>, <a class="txt" href="chord.html#E2:134"><i><font color="Green" title="E101">A2</font></i></a></i>;<br/><a NAME="E4:134_2_2"/><b>then </b>
<font color="Maroon" title="c12">vb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">b</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:134_2_2"><i><font color="Green" title="E118">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:134_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E22:134_2"><i><font color="Green" title="E115">A38</font></i></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_2_3"/>
( <font color="Maroon" title="c11">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">vb</font> or <font color="Maroon" title="c12">vb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">va</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E8:134_2"><i><font color="Green" title="E105">A28</font></i></a>, <a class="txt" href="chord.html#E19:134_2"><i><font color="Green" title="E113">A36</font></i></a>, <a class="txt" href="chord.html#E22:134_2"><i><font color="Green" title="E115">A38</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_3_1"><b>suppose </b></a><a NAME="E1:134_2_3_1"/>
<font color="Maroon" title="c11">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">vb</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:134_2_3_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b4">va</font>, <font color="Olive" title="b5">vb</font>, <font color="Olive" title="b6">vc</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b4">va</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b5">vb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b6">vc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">vb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">vc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">vb</font> &amp; <font color="Olive" title="b4">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">vc</font> &amp; not <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E8:134_2"><i><font color="Green" title="E105">A28</font></i></a>, <a class="txt" href="chord.html#E16:134_2"><i><font color="Green" title="E111">A34</font></i></a>, <a class="txt" href="chord.html#E4:134_2"><i><font color="Green" title="E103">A26</font></i></a>, <a class="txt" href="chord.html#E19:134_2"><i><font color="Green" title="E113">A36</font></i></a>, <a class="txt" href="chord.html#E22:134_2"><i><font color="Green" title="E115">A38</font></i></a>, <a class="txt" href="chord.html#E23:134_2"><i><font color="Green" title="E116">A39</font></i></a>, <a class="txt" href="chord.html#E24:134_2"><i><font color="Green" title="E117">A41</font></i></a>, <a class="txt" href="chord.html#E6:134_2"><i><font color="Green" title="E104">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_2_3_2"><b>suppose </b></a><a NAME="E1:134_2_3_2"/>
<font color="Maroon" title="c12">vb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">va</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:134_2_3_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b4">va</font>, <font color="Olive" title="b5">vb</font>, <font color="Olive" title="b6">vc</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b4">va</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b5">vb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b6">vc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">vb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">vc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">vb</font> &amp; <font color="Olive" title="b4">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">vc</font> &amp; not <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E8:134_2"><i><font color="Green" title="E105">A28</font></i></a>, <a class="txt" href="chord.html#E16:134_2"><i><font color="Green" title="E111">A34</font></i></a>, <a class="txt" href="chord.html#E4:134_2"><i><font color="Green" title="E103">A26</font></i></a>, <a class="txt" href="chord.html#E19:134_2"><i><font color="Green" title="E113">A36</font></i></a>, <a class="txt" href="chord.html#E22:134_2"><i><font color="Green" title="E115">A38</font></i></a>, <a class="txt" href="chord.html#E23:134_2"><i><font color="Green" title="E116">A39</font></i></a>, <a class="txt" href="chord.html#E24:134_2"><i><font color="Green" title="E117">A41</font></i></a>, <a class="txt" href="chord.html#E6:134_2"><i><font color="Green" title="E104">A27</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:134"/>
( <font color="Maroon" title="c2">V</font> is <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> iff for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/>for <font color="Olive" title="b4">va</font>, <font color="Olive" title="b5">vb</font>, <font color="Olive" title="b6">vc</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b4">va</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b5">vb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b6">vc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">vb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">vc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">vb</font> &amp; <font color="Olive" title="b4">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">vc</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )
 <b>by </b><i><a class="txt" href="chord.html#E3:134"><i><font color="Green" title="E102">A3</font></i></a></i>;<br/>


</div>
