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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <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 2 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  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="E1:64_1_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
( <font color="Maroon" title="c1">G</font> is 2 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  &amp; <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"> <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">G9</font> = <font color="Maroon" title="c1">G</font> as   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G9</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G9</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G9</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D1" onmouseover="rs('square_1/D1')" onmouseout="rh()">SQUARE_1:def 1</a></span>
<br/>.= 
1

;<br/>
<a NAME="E4:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c2">G9</font> <a href="glib_013.html#K2" title="GLIB_013:func.2">.size()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="glib_013.html#T12" onmouseover="rs('glib_013/T12')" onmouseout="rh()">GLIB_013:12</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_1_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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G9</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="c3">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G9</font><span class="kw"> such that </span><a NAME="E2:64_1_1_2"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c3">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/><a NAME="E3:64_1_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G9</font> <a href="glib_013.html#K2" title="GLIB_013:func.2">.size()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="glibpre1.html#T27" onmouseover="rs('glibpre1/T27')" onmouseout="rh()">GLIBPRE1:27</a>, <a class="ref" href="nat_1.html#T43" onmouseover="rs('nat_1/T43')" onmouseout="rh()">NAT_1:43</a></span>;<br/><a NAME="E4:64_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:64_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a>, <a class="ref" href="glib_001.html#D26" onmouseover="rs('glib_001/D26')" onmouseout="rh()">GLIB_001:def 26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:64_1_1_2"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c3">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c2">G9</font>
 <span class="kw">and </span><br/><a NAME="E8:64_1_1_2"/>
<font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G9</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<font color="Maroon" title="c4">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T28" onmouseover="rs('glibpre1/T28')" onmouseout="rh()">GLIBPRE1:28</a></span>;<br/><a NAME="E9:64_1_1_2"/>
<font color="Maroon" title="c3">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_2"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/><span class="kw">hence </span><a NAME="E10:64_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E7:64_1_1_2"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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="E6:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<font color="Maroon" title="c2">G9</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="64_1_1_3"><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">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G9</font> holds  <font color="Olive" title="b1">v</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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G9</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">v</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><br/><a NAME="E1:64_1_1_3"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">G9</font> <a href="glib_013.html#K2" title="GLIB_013:func.2">.size()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T43" onmouseover="rs('nat_1/T43')" onmouseout="rh()">NAT_1:43</a></span>;<br/><a NAME="E2:64_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">v</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="c2">G9</font> <a href="glib_013.html#K2" title="GLIB_013:func.2">.size()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="glib_000.html#T19" onmouseover="rs('glib_000/T19')" onmouseout="rh()">GLIB_000:19</a></span>;<br/><a NAME="E3:64_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">v</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> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_3"/>
<font color="Maroon" title="c3">v</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="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="E8:64_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E6:64_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="glpacy00.html#T9" target="_self" onmouseover="rs('glpacy00/T9')" onmouseout="rh()">Th17</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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