<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</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"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  iff  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span> )</span><br/>

<div><span class="kw">hereby </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"> (  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span> implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:208_1"/><span class="lab"><font color="Green" title="E125">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</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">  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span></span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="208_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">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> implies ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> ) ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> implies ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> ) ) &amp; ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> ) )</span><br/><div><span class="kw">hereby </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"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:208_1_1_1"/><span class="lab"><font color="Green" title="E126">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )</span><br/><a NAME="E2:208_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" onmouseover="rs('relat_1/T15')" onmouseout="rh()">RELAT_1:15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:208_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T160" target="_self" onmouseover="rs('glenum00/T160')" onmouseout="rh()">Th160</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"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">T1</font> = <font color="Maroon" title="c2">x</font> as   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T138" target="_self" onmouseover="rs('glenum00/T138')" onmouseout="rh()">Th138</a></span>;<br/><a NAME="E5:208_1_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:208_1_1_1"><span class="lab"><font color="Green" title="E126">A2</font></span></a>, <a class="ref" href="relat_1.html#T15" onmouseover="rs('relat_1/T15')" onmouseout="rh()">RELAT_1:15</a></span>;<br/><a NAME="E6:208_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T160" target="_self" onmouseover="rs('glenum00/T160')" onmouseout="rh()">Th160</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">T2</font> = <font color="Maroon" title="c3">y</font> as   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T138" target="_self" onmouseover="rs('glenum00/T138')" onmouseout="rh()">Th138</a></span>;<br/><a NAME="E8:208_1_1_1"/><span class="lab"><font color="Green" title="E127">A3</font></span>: 
<font color="Maroon" title="c5">T2</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:208_1"><span class="lab"><font color="Green" title="E125">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">T2</font><span class="kw"> such that </span><br/><a NAME="E10:208_1_1_1"/><span class="lab"><font color="Green" title="E128">A4</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">v</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="ref" href="glib_000.html#T22" onmouseover="rs('glib_000/T22')" onmouseout="rh()">GLIB_000:22</a></span>;<br/><a NAME="E11:208_1_1_1"/><span class="lab"><font color="Green" title="E129">A5</font></span>: 
<font color="Maroon" title="c4">T1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c5">T2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:208_1_1_1"><span class="lab"><font color="Green" title="E126">A2</font></span></a>, <a class="ref" href="glenum00.html#T159" target="_self" onmouseover="rs('glenum00/T159')" onmouseout="rh()">Th159</a></span>;<br/><a NAME="E12:208_1_1_1"/><span class="kw">then </span>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">T2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>;<br/><a NAME="E13:208_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A6</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">T2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:208_1_1_1"><span class="lab"><font color="Green" title="E128">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</a></span>;<br/><a NAME="E14:208_1_1_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:208_1_1_1"><span class="lab"><font color="Green" title="E127">A3</font></span></a></span>;<br/><a NAME="E15:208_1_1_1"/><span class="kw">then </span>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">T2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c4">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" onmouseover="rs('xboole_1/T2')" onmouseout="rh()">XBOOLE_1:2</a></span>;<br/><a NAME="E16:208_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">T2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:208_1_1_1"><span class="lab"><font color="Green" title="E130">A6</font></span></a>, <a class="ref" href="glib_000.html#T44" onmouseover="rs('glib_000/T44')" onmouseout="rh()">GLIB_000:44</a></span>;<br/><span class="kw">hence </span><a NAME="E17:208_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:208_1_1_1"><span class="lab"><font color="Green" title="E129">A5</font></span></a>, <a class="ref" href="glib_000.html#T87" onmouseover="rs('glib_000/T87')" onmouseout="rh()">GLIB_000:87</a>, <a class="ref" href="glib_009.html#T44" onmouseover="rs('glib_009/T44')" onmouseout="rh()">GLIB_009:44</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">assume </span><a NAME="E2:208_1_1"/><span class="lab"><font color="Green" title="E126">A7</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span><br/><a NAME="E3:208_1_1"/>
 <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D9" onmouseover="rs('relat_2/D9')" onmouseout="rh()">RELAT_2:def 9</a></span>;<br/><a NAME="E4:208_1_1"/><span class="kw">then </span>
 <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T160" target="_self" onmouseover="rs('glenum00/T160')" onmouseout="rh()">Th160</a></span>;<br/><span class="kw">hence </span><a NAME="E5:208_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:208_1_1"><span class="lab"><font color="Green" title="E126">A7</font></span></a>, <a class="ref" href="relat_2.html#D1" onmouseover="rs('relat_2/D1')" onmouseout="rh()">RELAT_2:def 1</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="E3:208_1"/>
 <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D10" onmouseover="rs('relat_1/D10')" onmouseout="rh()">RELAT_1:def 10</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">assume </span><a NAME="E2:208"/><span class="lab"><font color="Green" title="E125">A8</font></span>: 
 <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></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"> <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:208"/>
 not <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">G9</font> = <font color="Maroon" title="c1">G</font> as   non  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">e</font> =  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">v</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">w</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">V</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/>
<a NAME="E5:208"/><span class="lab"><font color="Green" title="E126">A9</font></span>: 
 the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>,<font color="Maroon" title="c2">G9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">V</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   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="c2">G9</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">S</font> =  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">T</font> =  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>;<br/>
<a NAME="E7:208"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T82" onmouseover="rs('funct_4/T82')" onmouseout="rh()">FUNCT_4:82</a></span>;<br/>
<a NAME="E8:208"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A10</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</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="ref" href="relat_1.html#T9" onmouseover="rs('relat_1/T9')" onmouseout="rh()">RELAT_1:9</a></span>;<br/>
<a NAME="E9:208"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T88" onmouseover="rs('funcop_1/T88')" onmouseout="rh()">FUNCOP_1:88</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">S</font> =  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>, <font color="Maroon" title="c11">T</font> =  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c7">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:208"><span class="lab"><font color="Green" title="E127">A10</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a>, <a class="ref" href="zfmisc_1.html#T7" onmouseover="rs('zfmisc_1/T7')" onmouseout="rh()">ZFMISC_1:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">H9</font> =  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>;<br/>
<a NAME="E11:208"/>
 not  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G9</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 
;<br/>
<a NAME="E12:208"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A11</font></span>: 
 <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T143" target="_self" onmouseover="rs('glenum00/T143')" onmouseout="rh()">Th143</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">H8</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E13:208"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</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="ref" href="glenum00.html#T13" target="_self" onmouseover="rs('glenum00/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E14:208"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A12</font></span>: 
 the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:208"/>
 the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>, <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:208"><span class="lab"><font color="Green" title="E129">A12</font></span></a>, <a class="txt" href="#E5:208"><span class="lab"><font color="Green" title="E126">A9</font></span></a>, <a class="ref" href="glib_000.html#T73" onmouseover="rs('glib_000/T73')" onmouseout="rh()">GLIB_000:73</a></span>;<br/>
<a NAME="E16:208"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A14</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/>
<a NAME="E17:208"/><span class="lab"><font color="Green" title="E131">A15</font></span>: 
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T43" onmouseover="rs('glib_000/T43')" onmouseout="rh()">GLIB_000:43</a></span>;<br/>
<a NAME="E18:208"/>
(  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  &amp;  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:208"><span class="lab"><font color="Green" title="E130">A14</font></span></a></span>;<br/>
<a NAME="E19:208"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A16</font></span>: 
(  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp;  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )
 
;<br/>
<a NAME="E20:208"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A17</font></span>: 
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:208"><span class="lab"><font color="Green" title="E131">A15</font></span></a></span>;<br/>
<a NAME="E21:208"/><span class="lab"><font color="Green" title="E134">A18</font></span>: 
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:208"><span class="lab"><font color="Green" title="E132">A16</font></span></a></span>;<br/>
<a NAME="E22:208"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"> the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <span class="p1">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G9</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">G9</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:208"><span class="lab"><font color="Green" title="E128">A11</font></span></a>, <a class="txt" href="#E20:208"><span class="lab"><font color="Green" title="E133">A17</font></span></a>, <a class="ref" href="glenum00.html#T159" target="_self" onmouseover="rs('glenum00/T159')" onmouseout="rh()">Th159</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:208"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:208"><span class="lab"><font color="Green" title="E134">A18</font></span></a>, <a class="txt" href="#E2:208"><span class="lab"><font color="Green" title="E125">A8</font></span></a>, <a class="ref" href="relat_1.html#D10" onmouseover="rs('relat_1/D10')" onmouseout="rh()">RELAT_1:def 10</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/>


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