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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">va</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">va</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E1:127"/><span class="lab"><font color="Green" title="E96">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" onmouseover="rs('chord/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E97">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>  &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>  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#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">va</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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/> not <font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> </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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">va</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E1:127_1"/><span class="lab"><font color="Green" title="E98">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="E2:127_1"/><span class="lab"><font color="Green" title="E99">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="E3:127_1"/><span class="lab"><font color="Green" title="E100">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="E4:127_1"/><span class="lab"><font color="Green" title="E101">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">va</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">va</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E5:127_1"/><span class="lab"><font color="Green" title="E102">A7</font></span>: 
<font color="Maroon" title="c6">va</font> <a href="tarski.html#R2" title="TARSKI: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="E6:127_1"/><span class="lab"><font color="Green" title="E103">A8</font></span>: 
<font color="Maroon" title="c7">vb</font> <a href="tarski.html#R2" title="TARSKI: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="E7:127_1"/><span class="lab"><font color="Green" title="E104">A9</font></span>: 
<font color="Maroon" title="c8">vc</font> <a href="tarski.html#R2" title="TARSKI: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="E8:127_1"/><span class="lab"><font color="Green" title="E105">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="E9:127_1"/><span class="lab"><font color="Green" title="E106">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="E10:127_1"/><span class="lab"><font color="Green" title="E107">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="E11:127_1"/><span class="lab"><font color="Green" title="E108">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="E12:127_1"/><span class="lab"><font color="Green" title="E109">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">set </span><font color="Maroon" title="c9">Gf</font> =  the   <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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font>;<br/><a NAME="E13:127_1"/><span class="lab"><font color="Green" title="E110">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="#E5:127_1"><span class="lab"><font color="Green" title="E102">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E14:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">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#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E15:127_1"/><span class="lab"><font color="Green" title="E112">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="#E5:127_1"><span class="lab"><font color="Green" title="E102">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E16:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:127_1"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="ref" href="finseq_6.html#D4" onmouseover="rs('finseq_6/D4')" onmouseout="rh()">FINSEQ_6:def 4</a></span>;<br/><a NAME="E17:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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="E18:127_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:127_1"><span class="lab"><font color="Green" title="E111">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E19:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A20</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI: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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E20:127_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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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#K5" title="NUMBERS:func.5">0</a> <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="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:127_1"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="txt" href="#E13:127_1"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="txt" href="#E17:127_1"><span class="lab"><font color="Green" title="E114">A19</font></span></a>, <a class="txt" href="#E14:127_1"><span class="lab"><font color="Green" title="E111">A16</font></span></a>, <a class="ref" href="finseq_6.html#D4" onmouseover="rs('finseq_6/D4')" onmouseout="rh()">FINSEQ_6:def 4</a></span>;<br/><a NAME="E21:127_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="tarski.html#R2" title="TARSKI: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="#E8:127_1"><span class="lab"><font color="Green" title="E105">A10</font></span></a>, <a class="txt" href="#E19:127_1"><span class="lab"><font color="Green" title="E115">A20</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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  the   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">jc</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E24:127_1"/><span class="lab"><font color="Green" title="E116">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="#E12:127_1"><span class="lab"><font color="Green" title="E109">A14</font></span></a>, <a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><a NAME="E25:127_1"/><span class="lab"><font color="Green" title="E117">A22</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="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#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><div><span class="lab"><font color="Green" title="E118">A23</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:127_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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:127_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#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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:127_1"><span class="lab"><font color="Green" title="E113">A18</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E3:127_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="#E24:127_1"><span class="lab"><font color="Green" title="E116">A21</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:127_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:127_1"><span class="lab"><font color="Green" title="E104">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E27:127_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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E28:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">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="tarski.html#R2" title="TARSKI: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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:127_1"><span class="lab"><font color="Green" title="E117">A22</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E29:127_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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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="#E15:127_1"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="txt" href="#E13:127_1"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="txt" href="#E26:127_1"><span class="lab"><font color="Green" title="E118">A23</font></span></a>, <a class="ref" href="finseq_6.html#D4" onmouseover="rs('finseq_6/D4')" onmouseout="rh()">FINSEQ_6:def 4</a></span>;<br/><a NAME="E30:127_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">c</font> <a href="tarski.html#R2" title="TARSKI: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="#E10:127_1"><span class="lab"><font color="Green" title="E107">A12</font></span></a>, <a class="txt" href="#E24:127_1"><span class="lab"><font color="Green" title="E116">A21</font></span></a>, <a class="txt" href="#E28:127_1"><span class="lab"><font color="Green" title="E119">A24</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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  the   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E32:127_1"/><span class="lab"><font color="Green" title="E120">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="#E15:127_1"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="txt" href="#E13:127_1"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="ref" href="chord.html#T107" target="_self" onmouseover="rs('chord/T107')" onmouseout="rh()">Th106</a></span>;<br/><a NAME="E33:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">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="#E3:127_1"><span class="lab"><font color="Green" title="E100">A5</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">jb</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E35:127_1"/><span class="lab"><font color="Green" title="E122">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="#E11:127_1"><span class="lab"><font color="Green" title="E108">A13</font></span></a>, <a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/><a NAME="E36:127_1"/><span class="lab"><font color="Green" title="E123">A28</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="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#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><div><span class="lab"><font color="Green" title="E124">A29</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:127_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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:127_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#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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:127_1"><span class="lab"><font color="Green" title="E113">A18</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E3:127_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="#E35:127_1"><span class="lab"><font color="Green" title="E122">A27</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:127_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:127_1"><span class="lab"><font color="Green" title="E103">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E38:127_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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E39:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">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="tarski.html#R2" title="TARSKI: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="finseq_6.html#K5" title="FINSEQ_6:func.5">-cut</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:127_1"><span class="lab"><font color="Green" title="E123">A28</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E40:127_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="finseq_6.html#K5" title="FINSEQ_6:func.5">-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="#E15:127_1"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="txt" href="#E13:127_1"><span class="lab"><font color="Green" title="E110">A15</font></span></a>, <a class="txt" href="#E37:127_1"><span class="lab"><font color="Green" title="E124">A29</font></span></a>, <a class="ref" href="finseq_6.html#D4" onmouseover="rs('finseq_6/D4')" onmouseout="rh()">FINSEQ_6:def 4</a></span>;<br/><a NAME="E41:127_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R2" title="TARSKI: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="#E9:127_1"><span class="lab"><font color="Green" title="E106">A11</font></span></a>, <a class="txt" href="#E35:127_1"><span class="lab"><font color="Green" title="E122">A27</font></span></a>, <a class="txt" href="#E39:127_1"><span class="lab"><font color="Green" title="E125">A30</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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  the   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E43:127_1"/><span class="lab"><font color="Green" title="E126">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="#E4:127_1"><span class="lab"><font color="Green" title="E101">A6</font></span></a>, <a class="txt" href="#E32:127_1"><span class="lab"><font color="Green" title="E120">A25</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>;<br/><span class="kw">assume </span><a NAME="E44:127_1"/>
<font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E45:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">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="#E8:127_1"><span class="lab"><font color="Green" title="E105">A10</font></span></a>, <a class="txt" href="#E15:127_1"><span class="lab"><font color="Green" title="E112">A17</font></span></a>, <a class="txt" href="#E13:127_1"><span class="lab"><font color="Green" title="E110">A15</font></span></a></span>;<br/><a NAME="E46:127_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="#E1:127"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E5:127_1"><span class="lab"><font color="Green" title="E102">A7</font></span></a>, <a class="txt" href="#E7:127_1"><span class="lab"><font color="Green" title="E104">A9</font></span></a>, <a class="txt" href="#E8:127_1"><span class="lab"><font color="Green" title="E105">A10</font></span></a>, <a class="txt" href="#E10:127_1"><span class="lab"><font color="Green" title="E107">A12</font></span></a>, <a class="txt" href="#E12:127_1"><span class="lab"><font color="Green" title="E109">A14</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E47:127_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A33</font></span>: 
<font color="Maroon" title="c13">cg</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <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> <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="#E33:127_1"><span class="lab"><font color="Green" title="E121">A26</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E48:127_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="#E2:127_1"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="txt" href="#E32:127_1"><span class="lab"><font color="Green" title="E120">A25</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>;<br/><a NAME="E49:127_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">bg</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <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> <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="#E33:127_1"><span class="lab"><font color="Green" title="E121">A26</font></span></a>, <a class="txt" href="#E43:127_1"><span class="lab"><font color="Green" title="E126">A31</font></span></a>, <a class="ref" href="chord.html#T52" target="_self" onmouseover="rs('chord/T52')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E50:127_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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>, the   <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1"><span class="lab"><font color="Green" title="E98">A3</font></span></a>, <a class="txt" href="#E45:127_1"><span class="lab"><font color="Green" title="E127">A32</font></span></a>, <a class="txt" href="#E47:127_1"><span class="lab"><font color="Green" title="E128">A33</font></span></a>, <a class="ref" href="chord.html#T68" target="_self" onmouseover="rs('chord/T68')" onmouseout="rh()">Th67</a></span>;<br/><span class="kw">hence </span><a NAME="E51:127_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:127_1"><span class="lab"><font color="Green" title="E126">A31</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:127"/><span class="lab"><font color="Green" title="E98">A34</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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" onmouseover="rs('chord/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>  implies  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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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></a><div class="add"><span class="kw">assume </span><a NAME="E1:127_2"/>
 not <font color="Maroon" title="c2">V</font> is  <a href="chord.html#V7" title="CHORD:attr.7">perfect</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="E2:127_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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>  ) )
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">n</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="E4:127_2"/><span class="lab"><font color="Green" title="E99">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="E5:127_2"/><span class="lab"><font color="Green" title="E100">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="E6:127_2"/><span class="lab"><font color="Green" title="E101">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="E7:127_2"/><span class="lab"><font color="Green" title="E102">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="#E4:127_2"><span class="lab"><font color="Green" title="E99">A35</font></span></a>, <a class="ref" href="chord.html#T107" target="_self" onmouseover="rs('chord/T107')" onmouseout="rh()">Th106</a></span>;<br/><a NAME="E8:127_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">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#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</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" onmouseover="rs('tarski/D3')" onmouseout="rh()">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="E11:127_2"/><span class="lab"><font color="Green" title="E104">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="E12:127_2"/><span class="lab"><font color="Green" title="E105">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="E13:127_2"/><span class="lab"><font color="Green" title="E106">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="E14:127_2"/><span class="lab"><font color="Green" title="E107">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="E15:127_2"/><span class="lab"><font color="Green" title="E108">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="E16:127_2"/><span class="lab"><font color="Green" title="E109">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="#E6:127_2"><span class="lab"><font color="Green" title="E101">A37</font></span></a>, <a class="ref" href="chord.html#T69" target="_self" onmouseover="rs('chord/T69')" onmouseout="rh()">Th68</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="#E8:127_2"><span class="lab"><font color="Green" title="E103">A39</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E18:127_2"/><span class="lab"><font color="Green" title="E110">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="#E15:127_2"><span class="lab"><font color="Green" title="E108">A44</font></span></a>, <a class="txt" href="#E7:127_2"><span class="lab"><font color="Green" title="E102">A38</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>;<br/><a NAME="E19:127_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" onmouseover="rs('chord/T1')" onmouseout="rh()">Th1</a></span>;<br/><a NAME="E20:127_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A47</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="tarski.html#R2" title="TARSKI: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="#E4:127_2"><span class="lab"><font color="Green" title="E99">A35</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E21:127_2"/>
<font color="Maroon" title="c8">b</font> <a href="tarski.html#R2" title="TARSKI: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="#E8:127_2"><span class="lab"><font color="Green" title="E103">A39</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E23:127_2"/><span class="lab"><font color="Green" title="E112">A48</font></span>: 
<font color="Maroon" title="c11">vb</font> <a href="tarski.html#R2" title="TARSKI: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="E24:127_2"/><span class="lab"><font color="Green" title="E113">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="#E3:127"><span class="lab"><font color="Green" title="E98">A34</font></span></a>, <a class="ref" href="finseq_2.html#T10" onmouseover="rs('finseq_2/T10')" onmouseout="rh()">FINSEQ_2:10</a></span>;<br/><a NAME="E25:127_2"/><span class="lab"><font color="Green" title="E114">A50</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E98">A34</font></span></a>, <a class="txt" href="#E8:127_2"><span class="lab"><font color="Green" title="E103">A39</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E115">A51</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:127_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:127_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">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="#E5:127_2"><span class="lab"><font color="Green" title="E100">A36</font></span></a>, <a class="txt" href="#E13:127_2"><span class="lab"><font color="Green" title="E106">A42</font></span></a>, <a class="txt" href="#E24:127_2"><span class="lab"><font color="Green" title="E113">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:127_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="#E1:127"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E20:127_2"><span class="lab"><font color="Green" title="E111">A47</font></span></a>, <a class="txt" href="#E8:127_2"><span class="lab"><font color="Green" title="E103">A39</font></span></a>, <a class="txt" href="#E25:127_2"><span class="lab"><font color="Green" title="E114">A50</font></span></a>, <a class="ref" href="chord.html#T16" target="_self" onmouseover="rs('chord/T16')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E4:127_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="#E2:127_2_1"><span class="lab"><font color="Green" title="E116">A52</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:127_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:127_2"><span class="lab"><font color="Green" title="E112">A48</font></span></a>, <a class="txt" href="#E24:127_2"><span class="lab"><font color="Green" title="E113">A49</font></span></a>, <a class="ref" href="finseq_4.html#T24" onmouseover="rs('finseq_4/T24')" onmouseout="rh()">FINSEQ_4:24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E27:127_2"/>
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI: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="#E8:127_2"><span class="lab"><font color="Green" title="E103">A39</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E29:127_2"/><span class="lab"><font color="Green" title="E116">A53</font></span>: 
<font color="Maroon" title="c12">va</font> <a href="tarski.html#R2" title="TARSKI: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="E30:127_2"/><span class="lab"><font color="Green" title="E117">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="#E3:127"><span class="lab"><font color="Green" title="E98">A34</font></span></a>, <a class="ref" href="finseq_2.html#T10" onmouseover="rs('finseq_2/T10')" onmouseout="rh()">FINSEQ_2:10</a></span>;<br/><a NAME="E31:127_2"/><span class="lab"><font color="Green" title="E118">A55</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E98">A34</font></span></a>, <a class="txt" href="#E8:127_2"><span class="lab"><font color="Green" title="E103">A39</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E119">A56</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="127_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:127_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:127_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">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="#E5:127_2"><span class="lab"><font color="Green" title="E100">A36</font></span></a>, <a class="txt" href="#E12:127_2"><span class="lab"><font color="Green" title="E105">A41</font></span></a>, <a class="txt" href="#E30:127_2"><span class="lab"><font color="Green" title="E117">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:127_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="#E1:127"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E20:127_2"><span class="lab"><font color="Green" title="E111">A47</font></span></a>, <a class="txt" href="#E8:127_2"><span class="lab"><font color="Green" title="E103">A39</font></span></a>, <a class="txt" href="#E31:127_2"><span class="lab"><font color="Green" title="E118">A55</font></span></a>, <a class="ref" href="chord.html#T16" target="_self" onmouseover="rs('chord/T16')" onmouseout="rh()">Th16</a></span>;<br/><a NAME="E4:127_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="#E2:127_2_2"><span class="lab"><font color="Green" title="E120">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:127_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:127_2"><span class="lab"><font color="Green" title="E116">A53</font></span></a>, <a class="txt" href="#E30:127_2"><span class="lab"><font color="Green" title="E117">A54</font></span></a>, <a class="ref" href="finseq_4.html#T24" onmouseover="rs('finseq_4/T24')" onmouseout="rh()">FINSEQ_4:24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E33:127_2"/><span class="lab"><font color="Green" title="E120">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="#E16:127_2"><span class="lab"><font color="Green" title="E109">A45</font></span></a>, <a class="txt" href="#E7:127_2"><span class="lab"><font color="Green" title="E102">A38</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>;<br/><a NAME="E34:127_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="#E14:127_2"><span class="lab"><font color="Green" title="E107">A43</font></span></a>, <a class="txt" href="#E7:127_2"><span class="lab"><font color="Green" title="E102">A38</font></span></a>, <a class="ref" href="chord.html#T45" target="_self" onmouseover="rs('chord/T45')" onmouseout="rh()">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E35:127_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="#E5:127_2"><span class="lab"><font color="Green" title="E100">A36</font></span></a>, <a class="txt" href="#E20:127_2"><span class="lab"><font color="Green" title="E111">A47</font></span></a>, <a class="txt" href="#E11:127_2"><span class="lab"><font color="Green" title="E104">A40</font></span></a>, <a class="txt" href="#E18:127_2"><span class="lab"><font color="Green" title="E110">A46</font></span></a>, <a class="txt" href="#E33:127_2"><span class="lab"><font color="Green" title="E120">A58</font></span></a>, <a class="txt" href="#E29:127_2"><span class="lab"><font color="Green" title="E116">A53</font></span></a>, <a class="txt" href="#E30:127_2"><span class="lab"><font color="Green" title="E117">A54</font></span></a>, <a class="txt" href="#E23:127_2"><span class="lab"><font color="Green" title="E112">A48</font></span></a>, <a class="txt" href="#E24:127_2"><span class="lab"><font color="Green" title="E113">A49</font></span></a>, <a class="txt" href="#E32:127_2"><span class="lab"><font color="Green" title="E119">A56</font></span></a>, <a class="txt" href="#E26:127_2"><span class="lab"><font color="Green" title="E115">A51</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:127"/>
( <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#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">va</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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="#E2:127"><span class="lab"><font color="Green" title="E97">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
