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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="115_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:115_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="115_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:115_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:115_1_1_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><span class="kw">hence </span><a NAME="E3:115_1_1_1_1_1"/>
<font color="Maroon" title="c3">H</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T95" target="_self" title="CHORD:th.95">Th95</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="115_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:115_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:115_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A1</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E91">A2</font></span>: 
<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
 <span class="kw">and </span><br/><a NAME="E2:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E92">A3</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="ref" href="glib_001.html#T176" title="GLIB_001:th.176">GLIB_001:176</a></span>;<br/><a NAME="E5:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E93">A4</font></span>: 
not <font color="Maroon" title="c6">P</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="ref" href="glib_001.html#T177" title="GLIB_001:th.177">GLIB_001:177</a></span>;<br/><a NAME="E6:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E94">A5</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E91">A2</font></span></a>, <a class="ref" href="glib_001.html#T115" title="GLIB_001:th.115">GLIB_001:115</a></span>;<br/><a NAME="E7:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E95">A6</font></span>: 
<font color="Maroon" title="c6">P</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="ref" href="glib_001.html#T177" title="GLIB_001:th.177">GLIB_001:177</a></span>;<br/><a NAME="E8:115_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></span>;<br/><a NAME="E9:115_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E94">A5</font></span></a>, <a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E11:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E96">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E12:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E97">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:115_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>
 <span class="kw">and </span><br/><a NAME="E14:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E98">A9</font></span>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="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>
 <span class="kw">and </span><br/><a NAME="E15:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E99">A10</font></span>: 
( <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> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T84" target="_self" title="CHORD:th.84">Th84</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:115_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E100">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E98">A9</font></span></a></span>;<br/><a NAME="E18:115_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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E19:115_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A12</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E96">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:115_1_1_1_1_2_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E21:115_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A13</font></span>: 
<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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E97">A8</font></span></a>, <a class="ref" href="glib_001.html#T8" title="GLIB_001:th.8">GLIB_001:8</a></span>;<br/><a NAME="E22:115_1_1_1_1_2_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E23:115_1_1_1_1_2_1"/><span class="kw">then </span>
<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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E97">A8</font></span></a>, <a class="txt" href="#E19:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E101">A12</font></span></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></span>;<br/><a NAME="E24:115_1_1_1_1_2_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1_1_1_2"><span class="lab"><font color="Green" title="E90">A1</font></span></a>, <a class="txt" href="#E17:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E100">A11</font></span></a>, <a class="txt" href="#E21:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E102">A13</font></span></a>, <a class="ref" href="chord.html#T19" target="_self" title="CHORD:th.19">Th19</a></span>;<br/><span class="kw">hence </span><a NAME="E25:115_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E92">A3</font></span></a>, <a class="txt" href="#E7:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E95">A6</font></span></a>, <a class="txt" href="#E5:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E93">A4</font></span></a>, <a class="txt" href="#E11:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E96">A7</font></span></a>, <a class="txt" href="#E12:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E97">A8</font></span></a>, <a class="txt" href="#E15:115_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E99">A10</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:115_1_1_1_1_2"/>
<font color="Maroon" title="c3">H</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#D11" target="_self" title="CHORD:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:115_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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