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<b>let </b><font color="Maroon" title="c3">H</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">H</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:116_1_1_1_1"/>
(  not <font color="Maroon" title="c2">V</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> or <font color="Maroon" title="c2">V</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:116_1_1_1_1_1"/>
<font color="Maroon" title="c2">V</font> is  not  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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">H</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a></span><br/><div class="add"><a NAME="E2:116_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">H</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E3:116_1_1_1_1_1"/>
<font color="Maroon" title="c3">H</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>
 <b>by </b><i><a class="ref" href="chord.html#T95" target="_self" title="CHORD:th.95">Th95</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:116_1_1_1_1_2"/>
<font color="Maroon" title="c2">V</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">H</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a></span><br/><div class="add"><a NAME="E2:116_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E90">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <font color="Maroon" title="c4">W</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies <font color="Maroon" title="c4">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:116_1_1_1_1_2_1"/><i><font color="Green" title="E91">A2</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 <b>and </b><br/><a NAME="E2:116_1_1_1_1_2_1"/><i><font color="Green" title="E92">A3</font></i>: 
<font color="Maroon" title="c4">W</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a></span><br/><a NAME="E3:116_1_1_1_1_2_1"/><i><font color="Green" title="E93">A4</font></i>: 
<font color="Maroon" title="c4">W</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="#E2:116_1_1_1_1_2_1"><i><font color="Green" title="E92">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">P</font> = <font color="Maroon" title="c4">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c5">P</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="#E3:116_1_1_1_1_2_1"><i><font color="Green" title="E93">A4</font></i></a>, <a class="ref" href="glib_001.html#T176" title="GLIB_001:th.176">GLIB_001:176</a></i>;<br/><a NAME="E6:116_1_1_1_1_2_1"/>
( <font color="Maroon" title="c4">W</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c4">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; <font color="Maroon" title="c4">W</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> )
 <b>by </b><i><a class="txt" href="#E2:116_1_1_1_1_2_1"><i><font color="Green" title="E92">A3</font></i></a></i>;<br/><a NAME="E7:116_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E94">A5</font></i>: 
( <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#T177" title="GLIB_001:th.177">GLIB_001:177</a></i>;<br/><a NAME="E8:116_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E95">A6</font></i>: 
<font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>;<br/><a NAME="E9:116_1_1_1_1_2_1"/>
<font color="Maroon" title="c6">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 <b>by </b><i><a class="txt" href="#E1:116_1_1_1_1_2_1"><i><font color="Green" title="E91">A2</font></i></a>, <a class="ref" href="glib_001.html#T115" title="GLIB_001:th.115">GLIB_001:115</a></i>;<br/><a NAME="E10:116_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="#E8:116_1_1_1_1_2_1"><i><font color="Green" title="E95">A6</font></i></a>, <a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">m</font>, <font color="Maroon" title="c8">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:116_1_1_1_1_2_1"/><i><font color="Green" title="E96">A7</font></i>: 
( <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c8">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="c6">P</font> )
 <b>and </b><br/><a NAME="E13:116_1_1_1_1_2_1"/>
<font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <b>and </b><br/><a NAME="E14:116_1_1_1_1_2_1"/><i><font color="Green" title="E97">A8</font></i>: 
(  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font> ) &amp; (  not <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font> ) ) ) )
 <b>by </b><i><a class="ref" href="chord.html#T84" target="_self" title="CHORD:th.84">Th84</a></i>;<br/><a NAME="E15:116_1_1_1_1_2_1"/><i><font color="Green" title="E98">A9</font></i>: 
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:116_1_1_1_1_2_1"/><i><font color="Green" title="E99">A10</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E14:116_1_1_1_1_2_1"><i><font color="Green" title="E97">A8</font></i></a></i>;<br/><a NAME="E18:116_1_1_1_1_2_1"/>
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E19:116_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="#E12:116_1_1_1_1_2_1"><i><font color="Green" title="E96">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:116_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="txt" href="#E12:116_1_1_1_1_2_1"><i><font color="Green" title="E96">A7</font></i></a>, <a class="txt" href="#E15:116_1_1_1_1_2_1"><i><font color="Green" title="E98">A9</font></i></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E21:116_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c4">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="txt" href="#E2:116_1_1_1_1_2"><i><font color="Green" title="E90">A1</font></i></a>, <a class="txt" href="#E17:116_1_1_1_1_2_1"><i><font color="Green" title="E99">A10</font></i></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E22:116_1_1_1_1_2_1"/>
<font color="Maroon" title="c4">W</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a>
 <b>by </b><i><a class="txt" href="#E3:116_1_1_1_1_2_1"><i><font color="Green" title="E93">A4</font></i></a>, <a class="txt" href="#E7:116_1_1_1_1_2_1"><i><font color="Green" title="E94">A5</font></i></a>, <a class="txt" href="#E12:116_1_1_1_1_2_1"><i><font color="Green" title="E96">A7</font></i></a>, <a class="txt" href="#E14:116_1_1_1_1_2_1"><i><font color="Green" title="E97">A8</font></i></a>, <a class="ref" href="chord.html#T85" target="_self" title="CHORD:th.85">Th85</a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:116_1_1_1_1_2"/>
<font color="Maroon" title="c3">H</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>
 <b>by </b><i><a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:116_1_1"/>
<font color="Maroon" title="c3">H</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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