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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being   <a href="chord.html#M3" title="CHORD:mode.3">VertexScheme</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a>  iff for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">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="b3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/>for <font color="Olive" title="b5">va</font>, <font color="Olive" title="b6">vb</font>, <font color="Olive" title="b7">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="b5">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="Olive" title="b1">V</font> &amp; <font color="Olive" title="b6">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="Olive" title="b1">V</font> &amp; <font color="Olive" title="b7">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="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">vb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">vc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b5">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">vb</font> &amp; <font color="Olive" title="b5">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">vc</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  )</span><br/>



<a NAME="E1:132"/><span class="lab"><font color="Green" title="E99">A1</font></span>: 
<font color="Maroon" title="c2">V</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D12" target="_self" title="CHORD:def.12">Def12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E100">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  implies for <font color="Olive" title="b1">va</font>, <font color="Olive" title="b2">vb</font>, <font color="Olive" title="b3">vc</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="b2">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="b3">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="b1">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="Olive" title="b2">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="Olive" title="b3">vc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Olive" title="b1">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">vb</font> &amp; <font color="Olive" title="b1">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">vc</font> holds <br/> not <font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:132_1"/><span class="lab"><font color="Green" title="E101">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:132_1"/><span class="lab"><font color="Green" title="E102">A4</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E3:132_1"/><span class="lab"><font color="Green" title="E103">A5</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E4:132_1"/><span class="lab"><font color="Green" title="E104">A6</font></span>: 
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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">va</font>, <font color="Olive" title="b2">vb</font>, <font color="Olive" title="b3">vc</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="b2">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="b3">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="b1">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="Olive" title="b2">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="Olive" title="b3">vc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Olive" title="b1">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">vb</font> &amp; <font color="Olive" title="b1">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">vc</font> holds <br/> not <font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies  not <font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:132_1"/><span class="lab"><font color="Green" title="E105">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:132_1"/><span class="lab"><font color="Green" title="E106">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:132_1"/><span class="lab"><font color="Green" title="E107">A9</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:132_1"/><span class="lab"><font color="Green" title="E108">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:132_1"/><span class="lab"><font color="Green" title="E109">A11</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E10:132_1"/><span class="lab"><font color="Green" title="E110">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E11:132_1"/><span class="lab"><font color="Green" title="E111">A13</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E12:132_1"/><span class="lab"><font color="Green" title="E112">A14</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> </span><br/><span class="kw">consider </span><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/><span class="kw">set </span><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="E14:132_1"/><span class="lab"><font color="Green" title="E113">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:132_1"><span class="lab"><font color="Green" title="E105">A7</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E15:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E16:132_1"/><span class="lab"><font color="Green" title="E115">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">va</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:132_1"><span class="lab"><font color="Green" title="E105">A7</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E17:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A18</font></span>: 
<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="nat_1.html#K2" title="NAT_1: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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">va</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:132_1"><span class="lab"><font color="Green" title="E113">A15</font></span></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><a NAME="E18:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A19</font></span>: 
 <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
 ;<br/><a NAME="E19:132_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:132_1"><span class="lab"><font color="Green" title="E114">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E20:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A20</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1: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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E21:132_1"/>
<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1: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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:132_1"><span class="lab"><font color="Green" title="E115">A17</font></span></a>, <a class="txt" href="#E14:132_1"><span class="lab"><font color="Green" title="E113">A15</font></span></a>, <a class="txt" href="#E18:132_1"><span class="lab"><font color="Green" title="E117">A19</font></span></a>, <a class="txt" href="#E15:132_1"><span class="lab"><font color="Green" title="E114">A16</font></span></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><a NAME="E22:132_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:132_1"><span class="lab"><font color="Green" title="E108">A10</font></span></a>, <a class="txt" href="#E20:132_1"><span class="lab"><font color="Green" title="E118">A20</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">jc</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E25:132_1"/><span class="lab"><font color="Green" title="E119">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:132_1"><span class="lab"><font color="Green" title="E112">A14</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E26:132_1"/><span class="lab"><font color="Green" title="E120">A22</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">jc</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><div><span class="lab"><font color="Green" title="E121">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:132_1_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:132_1_1"/><span class="kw">then </span>
<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#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:132_1"><span class="lab"><font color="Green" title="E116">A18</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:132_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:132_1"><span class="lab"><font color="Green" title="E119">A21</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:132_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:132_1"><span class="lab"><font color="Green" title="E107">A9</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E28:132_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">jc</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E29:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A24</font></span>: 
<font color="Maroon" title="c12">jc</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:132_1"><span class="lab"><font color="Green" title="E120">A22</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E30:132_1"/>
<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="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> <font color="Maroon" title="c12">jc</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:132_1"><span class="lab"><font color="Green" title="E115">A17</font></span></a>, <a class="txt" href="#E14:132_1"><span class="lab"><font color="Green" title="E113">A15</font></span></a>, <a class="txt" href="#E27:132_1"><span class="lab"><font color="Green" title="E121">A23</font></span></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><a NAME="E31:132_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:132_1"><span class="lab"><font color="Green" title="E110">A12</font></span></a>, <a class="txt" href="#E25:132_1"><span class="lab"><font color="Green" title="E119">A21</font></span></a>, <a class="txt" href="#E29:132_1"><span class="lab"><font color="Green" title="E122">A24</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><a NAME="E33:132_1"/><span class="lab"><font color="Green" title="E123">A25</font></span>: 
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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:132_1"><span class="lab"><font color="Green" title="E115">A17</font></span></a>, <a class="txt" href="#E14:132_1"><span class="lab"><font color="Green" title="E113">A15</font></span></a>, <a class="ref" href="chord.html#T107" target="_self" title="CHORD:th.107">Th107</a></span>;<br/><a NAME="E34:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A26</font></span>: 
<font color="Maroon" title="c11">ag</font>,<font color="Maroon" title="c13">cg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132_1"><span class="lab"><font color="Green" title="E103">A5</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">jb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E36:132_1"/><span class="lab"><font color="Green" title="E125">A27</font></span>: 
<font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">jb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:132_1"><span class="lab"><font color="Green" title="E111">A13</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E37:132_1"/><span class="lab"><font color="Green" title="E126">A28</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">jb</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><div><span class="lab"><font color="Green" title="E127">A29</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:132_1_2"/>
<font color="Maroon" title="c14">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:132_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">va</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">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#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:132_1"><span class="lab"><font color="Green" title="E116">A18</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:132_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:132_1"><span class="lab"><font color="Green" title="E125">A27</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:132_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:132_1"><span class="lab"><font color="Green" title="E106">A8</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E39:132_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">jb</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E40:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A30</font></span>: 
<font color="Maroon" title="c14">jb</font> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:132_1"><span class="lab"><font color="Green" title="E126">A28</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E41:132_1"/>
<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="c14">jb</font> <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> <font color="Maroon" title="c14">jb</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:132_1"><span class="lab"><font color="Green" title="E115">A17</font></span></a>, <a class="txt" href="#E14:132_1"><span class="lab"><font color="Green" title="E113">A15</font></span></a>, <a class="txt" href="#E38:132_1"><span class="lab"><font color="Green" title="E127">A29</font></span></a>, <a class="ref" href="graph_2.html#D1" title="GRAPH_2:def.1">GRAPH_2:def 1</a></span>;<br/><a NAME="E42:132_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:132_1"><span class="lab"><font color="Green" title="E109">A11</font></span></a>, <a class="txt" href="#E36:132_1"><span class="lab"><font color="Green" title="E125">A27</font></span></a>, <a class="txt" href="#E40:132_1"><span class="lab"><font color="Green" title="E128">A30</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><a NAME="E44:132_1"/><span class="lab"><font color="Green" title="E129">A31</font></span>: 
not <font color="Maroon" title="c15">bg</font>,<font color="Maroon" title="c13">cg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:132_1"><span class="lab"><font color="Green" title="E104">A6</font></span></a>, <a class="txt" href="#E33:132_1"><span class="lab"><font color="Green" title="E123">A25</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><span class="kw">assume </span><a NAME="E45:132_1"/>
<font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E46:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A32</font></span>: 
<font color="Maroon" title="c11">ag</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:132_1"><span class="lab"><font color="Green" title="E108">A10</font></span></a>, <a class="txt" href="#E16:132_1"><span class="lab"><font color="Green" title="E115">A17</font></span></a>, <a class="txt" href="#E14:132_1"><span class="lab"><font color="Green" title="E113">A15</font></span></a>, <a class="ref" href="chord.html#D13" target="_self" title="CHORD:def.13">Def13</a></span>;<br/><a NAME="E47:132_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E99">A1</font></span></a>, <a class="txt" href="#E5:132_1"><span class="lab"><font color="Green" title="E105">A7</font></span></a>, <a class="txt" href="#E7:132_1"><span class="lab"><font color="Green" title="E107">A9</font></span></a>, <a class="txt" href="#E8:132_1"><span class="lab"><font color="Green" title="E108">A10</font></span></a>, <a class="txt" href="#E10:132_1"><span class="lab"><font color="Green" title="E110">A12</font></span></a>, <a class="txt" href="#E12:132_1"><span class="lab"><font color="Green" title="E112">A14</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E48:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A33</font></span>: 
<font color="Maroon" title="c13">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="c11">ag</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:132_1"><span class="lab"><font color="Green" title="E124">A26</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><a NAME="E49:132_1"/>
<font color="Maroon" title="c11">ag</font>,<font color="Maroon" title="c15">bg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132_1"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="txt" href="#E33:132_1"><span class="lab"><font color="Green" title="E123">A25</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><a NAME="E50:132_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c11">ag</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:132_1"><span class="lab"><font color="Green" title="E124">A26</font></span></a>, <a class="txt" href="#E44:132_1"><span class="lab"><font color="Green" title="E129">A31</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" title="CHORD:th.52">Th52</a></span>;<br/><a NAME="E51:132_1"/><span class="kw">then </span>
 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="c15">bg</font>,<font color="Maroon" title="c13">cg</font>,<font color="Maroon" title="c9">Gf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132_1"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="txt" href="#E46:132_1"><span class="lab"><font color="Green" title="E130">A32</font></span></a>, <a class="txt" href="#E48:132_1"><span class="lab"><font color="Green" title="E131">A33</font></span></a>, <a class="ref" href="chord.html#T68" target="_self" title="CHORD:th.68">Th68</a></span>;<br/><span class="kw">hence </span><a NAME="E52:132_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:132_1"><span class="lab"><font color="Green" title="E129">A31</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:132"/><span class="lab"><font color="Green" title="E101">A34</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D12" target="_self" title="CHORD:def.12">Def12</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:132_2"/>
not <font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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>  )</span><br/><a NAME="E2:132_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D13" target="_self" title="CHORD:def.13">Def13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">n</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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><span class="kw"> such that </span><br/><a NAME="E4:132_2"/><span class="lab"><font color="Green" title="E102">A35</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:132_2"/><span class="lab"><font color="Green" title="E103">A36</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:132_2"/><span class="lab"><font color="Green" title="E104">A37</font></span>: 
not <font color="Maroon" title="c5">v</font> is  <a href="chord.html#V3" title="CHORD:attr.3">simplicial</a> 
 ;<br/><a NAME="E7:132_2"/><span class="lab"><font color="Green" title="E105">A38</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:132_2"><span class="lab"><font color="Green" title="E102">A35</font></span></a>, <a class="ref" href="chord.html#T107" target="_self" title="CHORD:th.107">Th107</a></span>;<br/><a NAME="E8:132_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A39</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">Gf</font><span class="kw"> such that </span><br/><a NAME="E11:132_2"/><span class="lab"><font color="Green" title="E107">A40</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">and </span><br/><a NAME="E12:132_2"/><span class="lab"><font color="Green" title="E108">A41</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">and </span><br/><a NAME="E13:132_2"/><span class="lab"><font color="Green" title="E109">A42</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">and </span><br/><a NAME="E14:132_2"/><span class="lab"><font color="Green" title="E110">A43</font></span>: 
<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c7">a</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">and </span><br/><a NAME="E15:132_2"/><span class="lab"><font color="Green" title="E111">A44</font></span>: 
<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c8">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">and </span><br/><a NAME="E16:132_2"/><span class="lab"><font color="Green" title="E112">A45</font></span>: 
not <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:132_2"><span class="lab"><font color="Green" title="E104">A37</font></span></a>, <a class="ref" href="chord.html#T69" target="_self" title="CHORD:th.69">Th69</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">ag</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">bg</font> = <font color="Maroon" title="c8">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:132_2"><span class="lab"><font color="Green" title="E106">A39</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E18:132_2"/><span class="lab"><font color="Green" title="E113">A46</font></span>: 
<font color="Maroon" title="c6">vg</font>,<font color="Maroon" title="c10">bg</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:132_2"><span class="lab"><font color="Green" title="E111">A44</font></span></a>, <a class="txt" href="#E7:132_2"><span class="lab"><font color="Green" title="E105">A38</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><a NAME="E19:132_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T1" target="_self" title="CHORD:th.1">Th1</a></span>;<br/><a NAME="E20:132_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A47</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:132_2"><span class="lab"><font color="Green" title="E102">A35</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E21:132_2"/>
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:132_2"><span class="lab"><font color="Green" title="E106">A39</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">vb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E23:132_2"/><span class="lab"><font color="Green" title="E115">A48</font></span>: 
<font color="Maroon" title="c11">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>
 <span class="kw">and </span><br/><a NAME="E24:132_2"/><span class="lab"><font color="Green" title="E116">A49</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">vb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132"><span class="lab"><font color="Green" title="E101">A34</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><a NAME="E25:132_2"/><span class="lab"><font color="Green" title="E117">A50</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132"><span class="lab"><font color="Green" title="E101">A34</font></span></a>, <a class="txt" href="#E8:132_2"><span class="lab"><font color="Green" title="E106">A39</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E118">A51</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:132_2_1"/>
<font color="Maroon" title="c11">vb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:132_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A52</font></span>: 
<font color="Maroon" title="c11">vb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:132_2"><span class="lab"><font color="Green" title="E103">A36</font></span></a>, <a class="txt" href="#E13:132_2"><span class="lab"><font color="Green" title="E109">A42</font></span></a>, <a class="txt" href="#E24:132_2"><span class="lab"><font color="Green" title="E116">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:132_2_1"/>
<font color="Maroon" title="c8">b</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E99">A1</font></span></a>, <a class="txt" href="#E20:132_2"><span class="lab"><font color="Green" title="E114">A47</font></span></a>, <a class="txt" href="#E8:132_2"><span class="lab"><font color="Green" title="E106">A39</font></span></a>, <a class="txt" href="#E25:132_2"><span class="lab"><font color="Green" title="E117">A50</font></span></a>, <a class="ref" href="chord.html#T16" target="_self" title="CHORD:th.16">Th16</a></span>;<br/><a NAME="E4:132_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">vb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">b</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132_2_1"><span class="lab"><font color="Green" title="E119">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:132_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:132_2"><span class="lab"><font color="Green" title="E115">A48</font></span></a>, <a class="txt" href="#E24:132_2"><span class="lab"><font color="Green" title="E116">A49</font></span></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E27:132_2"/>
<font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:132_2"><span class="lab"><font color="Green" title="E106">A39</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">va</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E29:132_2"/><span class="lab"><font color="Green" title="E119">A53</font></span>: 
<font color="Maroon" title="c12">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>
 <span class="kw">and </span><br/><a NAME="E30:132_2"/><span class="lab"><font color="Green" title="E120">A54</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">va</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132"><span class="lab"><font color="Green" title="E101">A34</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><a NAME="E31:132_2"/><span class="lab"><font color="Green" title="E121">A55</font></span>: 
<font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:132"><span class="lab"><font color="Green" title="E101">A34</font></span></a>, <a class="txt" href="#E8:132_2"><span class="lab"><font color="Green" title="E106">A39</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E122">A56</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:132_2_2"/>
<font color="Maroon" title="c12">va</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:132_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A57</font></span>: 
<font color="Maroon" title="c12">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:132_2"><span class="lab"><font color="Green" title="E103">A36</font></span></a>, <a class="txt" href="#E12:132_2"><span class="lab"><font color="Green" title="E108">A41</font></span></a>, <a class="txt" href="#E30:132_2"><span class="lab"><font color="Green" title="E120">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:132_2_2"/>
<font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E99">A1</font></span></a>, <a class="txt" href="#E20:132_2"><span class="lab"><font color="Green" title="E114">A47</font></span></a>, <a class="txt" href="#E8:132_2"><span class="lab"><font color="Green" title="E106">A39</font></span></a>, <a class="txt" href="#E31:132_2"><span class="lab"><font color="Green" title="E121">A55</font></span></a>, <a class="ref" href="chord.html#T16" target="_self" title="CHORD:th.16">Th16</a></span>;<br/><a NAME="E4:132_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">va</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">a</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132_2_2"><span class="lab"><font color="Green" title="E123">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:132_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:132_2"><span class="lab"><font color="Green" title="E119">A53</font></span></a>, <a class="txt" href="#E30:132_2"><span class="lab"><font color="Green" title="E120">A54</font></span></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E33:132_2"/><span class="lab"><font color="Green" title="E123">A58</font></span>: 
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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:132_2"><span class="lab"><font color="Green" title="E112">A45</font></span></a>, <a class="txt" href="#E7:132_2"><span class="lab"><font color="Green" title="E105">A38</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><a NAME="E34:132_2"/>
<font color="Maroon" title="c6">vg</font>,<font color="Maroon" title="c9">ag</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:132_2"><span class="lab"><font color="Green" title="E110">A43</font></span></a>, <a class="txt" href="#E7:132_2"><span class="lab"><font color="Green" title="E105">A38</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" title="CHORD:th.45">Th45</a></span>;<br/><span class="kw">hence </span><a NAME="E35:132_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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:132_2"><span class="lab"><font color="Green" title="E103">A36</font></span></a>, <a class="txt" href="#E20:132_2"><span class="lab"><font color="Green" title="E114">A47</font></span></a>, <a class="txt" href="#E11:132_2"><span class="lab"><font color="Green" title="E107">A40</font></span></a>, <a class="txt" href="#E18:132_2"><span class="lab"><font color="Green" title="E113">A46</font></span></a>, <a class="txt" href="#E33:132_2"><span class="lab"><font color="Green" title="E123">A58</font></span></a>, <a class="txt" href="#E29:132_2"><span class="lab"><font color="Green" title="E119">A53</font></span></a>, <a class="txt" href="#E30:132_2"><span class="lab"><font color="Green" title="E120">A54</font></span></a>, <a class="txt" href="#E23:132_2"><span class="lab"><font color="Green" title="E115">A48</font></span></a>, <a class="txt" href="#E24:132_2"><span class="lab"><font color="Green" title="E116">A49</font></span></a>, <a class="txt" href="#E32:132_2"><span class="lab"><font color="Green" title="E122">A56</font></span></a>, <a class="txt" href="#E26:132_2"><span class="lab"><font color="Green" title="E118">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:132"/>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132"><span class="lab"><font color="Green" title="E100">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
