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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</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_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glib_013.html#K12" title="GLIB_013:func.12">.minDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c1">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:63"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glib_013.html#K12" title="GLIB_013:func.12">.minDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <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="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_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"> <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1"/>
 not <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</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">consider </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:63_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds  not <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">C1</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">C2</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span>;<br/><a NAME="E4:63_1"/>
 not <font color="Maroon" title="c3">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="glib_002.html#D5" onmouseover="rs('glib_002/D5')" onmouseout="rh()">GLIB_002:def 5</a></span>;<br/><a NAME="E5:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_008.html#T22" onmouseover="rs('glib_008/T22')" onmouseout="rh()">GLIB_008:22</a></span>;<br/><a NAME="E6:63_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E7:63_1"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">v9</font> = <font color="Maroon" title="c2">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="glib_002.html#T9" onmouseover="rs('glib_002/T9')" onmouseout="rh()">GLIB_002:9</a></span>;<br/><a NAME="E9:63_1"/>
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="glib_013.html#K12" title="GLIB_013:func.12">.minDegree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="glpacy00.html#E13"><span class="lab"><font color="Green" title="E13">Th16</font></span></a></span>;<br/><a NAME="E10:63_1"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="glib_013.html#K12" title="GLIB_013:func.12">.minDegree()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E11:63_1"/><span class="kw">then </span>
 not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 ;<br/><a NAME="E12:63_1"/><span class="kw">then </span>
(  not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:63_1"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
( <font color="Maroon" title="c7">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">v2</font> &amp; <font color="Maroon" title="c7">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c8">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 <span class="kw">and </span><br/><a NAME="E15:63_1"/>
<font color="Maroon" title="c8">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="glib_002.html#T43" onmouseover="rs('glib_002/T43')" onmouseout="rh()">GLIB_002:43</a></span>;<br/><a NAME="E16:63_1"/>
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="glib_013.html#K12" title="GLIB_013:func.12">.minDegree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="glpacy00.html#E13"><span class="lab"><font color="Green" title="E13">Th16</font></span></a></span>;<br/><a NAME="E17:63_1"/><span class="kw">then </span>
 the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="glib_013.html#K12" title="GLIB_013:func.12">.minDegree()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E18:63_1"/><span class="kw">then </span>
 not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 ;<br/><a NAME="E19:63_1"/><span class="kw">then </span>
(  not  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">w1</font>, <font color="Maroon" title="c10">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E21:63_1"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
( <font color="Maroon" title="c9">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">w2</font> &amp; <font color="Maroon" title="c9">w1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c10">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 <span class="kw">and </span><br/><a NAME="E22:63_1"/>
<font color="Maroon" title="c10">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="glib_002.html#T43" onmouseover="rs('glib_002/T43')" onmouseout="rh()">GLIB_002:43</a></span>;<br/><a NAME="E23:63_1"/>
( <font color="Maroon" title="c7">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> &amp; <font color="Maroon" title="c8">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:63_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a></span>;<br/><a NAME="E24:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c8">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a></span>;<br/><a NAME="E25:63_1"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">w1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:63_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E26:63_1"/>
(  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T8" target="_self" onmouseover="rs('glpacy00/T8')" onmouseout="rh()">Th12</a></span>;<br/><a NAME="E27:63_1"/><span class="kw">then </span>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c8">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">w1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:63_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E25:63_1"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E28:63_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c8">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">w1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><a NAME="E29:63_1"/><span class="kw">then </span>
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c8">v2</font>,<font color="Maroon" title="c9">w1</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T3" onmouseover="rs('enumset1/T3')" onmouseout="rh()">ENUMSET1:3</a></span>;<br/><a NAME="E30:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A10</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c7">v1</font>,<font color="Maroon" title="c8">v2</font>,<font color="Maroon" title="c9">w1</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_1:11</a></span>;<br/><a NAME="E31:63_1"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<font color="Maroon" title="c7">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E6:63_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E7:63_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><a NAME="E32:63_1"/>
<font color="Maroon" title="c8">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E6:63_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E7:63_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/><a NAME="E33:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A12</font></span>: 
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E14:63_1"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E30:63_1"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E31:63_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="card_2.html#T58" onmouseover="rs('card_2/T58')" onmouseout="rh()">CARD_2:58</a></span>;<br/><a NAME="E34:63_1"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><a NAME="E35:63_1"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:63_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E36:63_1"/>
contradiction
 ; <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="E3:63"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">GLPACY00:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">F</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">assume </span><a NAME="E4:63"/>
 not <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:63"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" onmouseover="rs('ordinal1/T16')" onmouseout="rh()">ORDINAL1:16</a>, <a class="ref" href="ordinal1.html#T21" onmouseover="rs('ordinal1/T21')" onmouseout="rh()">ORDINAL1:21</a></span>;<br/>
<a NAME="E6:63"/><span class="kw">then </span>
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K5" title="GLIB_002:func.5">.numComponents()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="glpacy00.html#T3" target="_self" onmouseover="rs('glpacy00/T3')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E7:63"/><span class="kw">then </span>
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D9" onmouseover="rs('glib_002/D9')" onmouseout="rh()">GLIB_002:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">c2</font>, <font color="Maroon" title="c6">c3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:63"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
( <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Maroon" title="c5">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  )
 <span class="kw">and </span><br/><a NAME="E10:63"/><span class="lab"><font color="Green" title="E42">A15</font></span>: 
( <font color="Maroon" title="c6">c3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Maroon" title="c4">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c2</font> &amp; <font color="Maroon" title="c4">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c3</font> &amp; <font color="Maroon" title="c5">c2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c3</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pencil_1.html#T5" onmouseover="rs('pencil_1/T5')" onmouseout="rh()">PENCIL_1:5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font><span class="kw"> such that </span><br/><a NAME="E12:63"/><span class="lab"><font color="Green" title="E43">A16</font></span>: 
<font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:63"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="glib_002.html#D8" onmouseover="rs('glib_002/D8')" onmouseout="rh()">GLIB_002:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font><span class="kw"> such that </span><br/><a NAME="E14:63"/><span class="lab"><font color="Green" title="E44">A17</font></span>: 
<font color="Maroon" title="c5">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:63"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="glib_002.html#D8" onmouseover="rs('glib_002/D8')" onmouseout="rh()">GLIB_002:def 8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">v3</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font><span class="kw"> such that </span><br/><a NAME="E16:63"/><span class="lab"><font color="Green" title="E45">A18</font></span>: 
<font color="Maroon" title="c6">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="glib_002.html#D8" onmouseover="rs('glib_002/D8')" onmouseout="rh()">GLIB_002:def 8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">C1</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">v1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">C2</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">C3</font> =  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">v3</font></span>)</span>;<br/>
<a NAME="E17:63"/><span class="lab"><font color="Green" title="E46">A19</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E18:63"/><span class="lab"><font color="Green" title="E47">A20</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<a NAME="E19:63"/><span class="lab"><font color="Green" title="E48">A21</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">v3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">v3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E21:63"/><span class="lab"><font color="Green" title="E49">A22</font></span>: 
( <font color="Maroon" title="c13">w1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c13">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">v1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="glpacy00.html#T4" target="_self" onmouseover="rs('glpacy00/T4')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E23:63"/><span class="lab"><font color="Green" title="E50">A23</font></span>: 
( <font color="Maroon" title="c14">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c14">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="glpacy00.html#T4" target="_self" onmouseover="rs('glpacy00/T4')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">w3</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E25:63"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
( <font color="Maroon" title="c15">w3</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c15">w3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">v3</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="glpacy00.html#T4" target="_self" onmouseover="rs('glpacy00/T4')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E26:63"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<font color="Maroon" title="c13">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">w2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c16">u1</font> = <font color="Maroon" title="c13">w1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:63"><span class="lab"><font color="Green" title="E49">A22</font></span></a></span>;<br/>
<a NAME="E2:63_2"/>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c16">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:63"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E21:63"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="glib_002.html#T12" onmouseover="rs('glib_002/T12')" onmouseout="rh()">GLIB_002:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:63_2"/>
<font color="Maroon" title="c13">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E12:63"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E14:63"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E18:63"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E23:63"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="glib_002.html#T12" onmouseover="rs('glib_002/T12')" onmouseout="rh()">GLIB_002:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:63"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<font color="Maroon" title="c14">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">w3</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c16">u2</font> = <font color="Maroon" title="c14">w2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:63"><span class="lab"><font color="Green" title="E50">A23</font></span></a></span>;<br/>
<a NAME="E2:63_3"/>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c16">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:63"><span class="lab"><font color="Green" title="E47">A20</font></span></a>, <a class="txt" href="#E23:63"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="glib_002.html#T12" onmouseover="rs('glib_002/T12')" onmouseout="rh()">GLIB_002:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:63_3"/>
<font color="Maroon" title="c14">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">w3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E14:63"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E16:63"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E19:63"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E25:63"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="glib_002.html#T12" onmouseover="rs('glib_002/T12')" onmouseout="rh()">GLIB_002:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E28:63"/>
<font color="Maroon" title="c15">w3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">w1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="63_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c16">u3</font> = <font color="Maroon" title="c15">w3</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:63"><span class="lab"><font color="Green" title="E51">A24</font></span></a></span>;<br/>
<a NAME="E2:63_4"/>
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c9">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c16">u3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63"><span class="lab"><font color="Green" title="E48">A21</font></span></a>, <a class="txt" href="#E25:63"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="glib_002.html#T12" onmouseover="rs('glib_002/T12')" onmouseout="rh()">GLIB_002:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:63_4"/>
<font color="Maroon" title="c15">w3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="txt" href="#E12:63"><span class="lab"><font color="Green" title="E43">A16</font></span></a>, <a class="txt" href="#E16:63"><span class="lab"><font color="Green" title="E45">A18</font></span></a>, <a class="txt" href="#E17:63"><span class="lab"><font color="Green" title="E46">A19</font></span></a>, <a class="txt" href="#E21:63"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="glib_002.html#T12" onmouseover="rs('glib_002/T12')" onmouseout="rh()">GLIB_002:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A27</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c13">w1</font>,<font color="Maroon" title="c14">w2</font>,<font color="Maroon" title="c15">w3</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:63"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E27:63"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="card_2.html#T58" onmouseover="rs('card_2/T58')" onmouseout="rh()">CARD_2:58</a></span>;<br/>
<a NAME="E30:63"/>
( <font color="Maroon" title="c13">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c14">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c15">w3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:63"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E23:63"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E25:63"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="glib_006.html#D8" onmouseover="rs('glib_006/D8')" onmouseout="rh()">GLIB_006:def 8</a></span>;<br/>
<a NAME="E31:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A28</font></span>: 
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:63"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_1:11</a>, <a class="ref" href="zfmisc_1.html#T133" onmouseover="rs('zfmisc_1/T133')" onmouseout="rh()">ZFMISC_1:133</a></span>;<br/>
<a NAME="E32:63"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E33:63"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E31:63"><span class="lab"><font color="Green" title="E55">A28</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:63"/>
contradiction
 ; <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>
