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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>   <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b2">P</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">v</font> holds <br/> ( <font color="Olive" title="b2">P</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>  &amp; <font color="Olive" title="b2">P</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</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">C</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">  for <font color="Olive" title="b1">P</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">v</font> holds <br/> ( <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">v</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="c3">P</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>  &amp; <font color="Maroon" title="c3">P</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</span><br/>





<span class="kw">thus </span><a NAME="E1:110"/>
<font color="Maroon" title="c3">P</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</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="c3">P</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> </span><br/>

<a NAME="E2:110"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">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#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">P</font>  holds  not <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c4">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><a NAME="E2:110_1"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c4">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 ; <a class="txt" 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">reconsider </span><font color="Maroon" title="c5">W9</font> = <font color="Maroon" title="c4">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" onmouseover="rs('glib_001/T167')" onmouseout="rh()">GLIB_001:167</a></span>;<br/><a NAME="E4:110_1"/>
<font color="Maroon" title="c5">W9</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="ref" href="glib_006.html#T24" onmouseover="rs('glib_006/T24')" onmouseout="rh()">GLIB_006:24</a></span>;<br/><span class="kw">then </span> <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">C</font> = 
<font color="Maroon" title="c5">W9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T31" target="_self" onmouseover="rs('glpacy00/T31')" onmouseout="rh()">Th39</a></span>
<br/>.= 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>
;<br/><a NAME="E6:110_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><span class="kw">hence </span><a NAME="E7:110_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</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="E4:110"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c3">P</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">P</font>  ex <font color="Olive" title="b3">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">P</font> st <font color="Olive" title="b3">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">w</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">w</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">P</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">  ex <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="c3">P</font> st <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="c4">u</font>,<font color="Maroon" title="c5">w</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c6">C9</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E2:110_2"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c6">C9</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 2</a></span>;<br/><a NAME="E3:110_2"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">C</font>
 ;<br/><a NAME="E4:110_2"/><span class="kw">then </span>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_2"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="glpacy00.html#T31" target="_self" onmouseover="rs('glpacy00/T31')" onmouseout="rh()">Th39</a></span>;<br/><a NAME="E5:110_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A5</font></span>: 
( <font color="Maroon" title="c4">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">C9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c5">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">C9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E6:110_2"/>
<font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E7:110_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">W</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E9:110_2"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
( <font color="Maroon" title="c7">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">w</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_2"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E5:110_2"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="glpacy00.html#E3"><span class="lab"><font color="Green" title="E3">Th5</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">W</font> = <font color="Maroon" title="c7">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:110_2"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="ref" href="glib_001.html#T171" onmouseover="rs('glib_001/T171')" onmouseout="rh()">GLIB_001:171</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">W</font> = <font color="Maroon" title="c8">W</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="c9">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">w</font></span><br/><span class="kw">thus </span><a NAME="E11:110_2"/>
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:110_2"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</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="E6:110"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<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">hence </span><a NAME="E7:110"/>
<font color="Maroon" title="c3">P</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="#E4:110"><span class="lab"><font color="Green" title="E54">A3</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="c3">P</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="c4">u</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">u</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">reconsider </span><font color="Maroon" title="c5">w</font> = <font color="Maroon" title="c4">u</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:110"/>
<font color="Maroon" title="c4">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">w</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="glib_000.html#T81" onmouseover="rs('glib_000/T81')" onmouseout="rh()">GLIB_000:81</a></span>;<br/>
<a NAME="E10:110"/><span class="kw">then </span>
<font color="Maroon" title="c4">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_016.html#D4" onmouseover="rs('glib_016/D4')" onmouseout="rh()">GLIB_016:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:110"/>
<font color="Maroon" title="c4">u</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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#T3" onmouseover="rs('field_5/T3')" onmouseout="rh()">FIELD_5:3</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>
