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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G1</font> be   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" 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="c1">G1</font><br/>  for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">G2</font> being   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b2">e</font>  st <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/><font color="Olive" title="b3">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">G2</font> being   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">e</font>  st <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/><font color="Olive" title="b2">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">e</font>  st <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/><font color="Olive" title="b1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">G2</font> be   <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">e</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  implies <font color="Maroon" title="c4">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:63"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">and </span><br/><a NAME="E2:63"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font>, <font color="Maroon" title="c6">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<a NAME="E4:63"/>
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">X</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E6:63"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 <span class="kw">and </span><br/><a NAME="E7:63"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="glib_001.html#T157" title="GLIB_001:th.157">GLIB_001:157</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">X2</font> = <font color="Maroon" title="c7">X</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></span>;<br/>
<a NAME="E9:63"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c8">X2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/>
<a NAME="E10:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c8">X2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c5">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T23" title="GLIB_001:th.23">GLIB_001:23</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c10">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font>; <a class="txt" 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font></span><br/><a NAME="E1:63_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">G2</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">G1</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">u9</font> = <font color="Maroon" title="c9">u</font>, <font color="Maroon" title="c12">v9</font> = <font color="Maroon" title="c10">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">C</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E4:63_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c13">C</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c11">u9</font>,<font color="Maroon" title="c12">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font>;<br/><a NAME="E5:63_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c11">u9</font>,<font color="Maroon" title="c12">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="glib_001.html#T160" title="GLIB_001:th.160">GLIB_001:160</a></span>;<br/><a NAME="E6:63_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A9</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E7:63_1"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_1_1"/>
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  or  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c17">m</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:63_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">and </span><br/><a NAME="E5:63_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:63_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:63_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">P1</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c17">m</font>);<br/><span class="kw">set </span><font color="Maroon" title="c19">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span>);<br/><a NAME="E8:63_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <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 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E9:63_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A17</font></span>: 
<font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_1_1_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c17">m</font>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" 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="c20">x</font> being   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:63_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c17">m</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:63_1_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A20</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c17">m</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></span>;<br/><a NAME="E6:63_1_1_1_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E4:63_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="glib_001.html#T45" title="GLIB_001:th.45">GLIB_001:45</a></span>;<br/><a NAME="E7:63_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:63_1_1_1_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c17">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E4:63_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:63_1_1_1_1_1"/><span class="kw">then </span>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:63_1_1_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E5:63_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A20</font></span></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></span>;<br/><span class="kw">hence </span><a NAME="E10:63_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E9:63_1_1_1_1"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="txt" href="#E3:63_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E7:63_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="glib_001.html#T138" title="GLIB_001:th.138">GLIB_001:138</a></span>; <a class="txt" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">P19</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c17">m</font>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:63_1_1_1_1_2"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" 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="c21">x</font> being   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:63_1_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A22</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">x</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A23</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:63_1_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A24</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T99" title="GLIB_001:th.99">GLIB_001:99</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">x1</font> = <font color="Maroon" title="c21">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A22</font></span></a>, <a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E7:63_1_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A25</font></span>: 
<font color="Maroon" title="c22">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1_1_1_1_2"><span class="lab"><font color="Green" title="E60">A23</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E8:63_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><a NAME="E9:63_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E10:63_1_1_1_1_2"/>
<font color="Maroon" title="c22">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font>
 ;<br/><a NAME="E11:63_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A27</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1_1_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E5:63_1_1_1_1_2"><span class="lab"><font color="Green" title="E61">A24</font></span></a>, <a class="txt" href="#E7:63_1_1_1_1_2"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="ref" href="glib_001.html#T36" title="GLIB_001:th.36">GLIB_001:36</a></span>;<br/><a NAME="E12:63_1_1_1_1_2"/>
<font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:63_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E14:63_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E15:63_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1_1_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="txt" href="#E11:63_1_1_1_1_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="txt" href="#E9:63_1_1_1_1_2"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E13:63_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="ref" href="glib_001.html#T138" title="GLIB_001:th.138">GLIB_001:138</a></span>; <a class="txt" 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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">P29</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">a</font> = <font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c23">b</font> = <font color="Maroon" title="c16">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T51" title="GLIB_000:th.51">GLIB_000:51</a></span>;<br/><a NAME="E15:63_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E16:63_1_1_1_1"/><span class="kw">then </span>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (1,<font color="Maroon" title="c17">m</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c22">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E4:63_1_1_1_1"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="txt" href="#E8:63_1_1_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a>, <a class="ref" href="jordan12.html#T2" title="JORDAN12:th.2">JORDAN12:2</a></span>;<br/><a NAME="E17:63_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A29</font></span>: 
<font color="Maroon" title="c20">P19</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c22">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><a NAME="E18:63_1_1_1_1"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:63_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="txt" href="#E3:63_1_1_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="txt" href="#E6:63_1_1_1_1"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/><a NAME="E19:63_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A30</font></span>: 
<font color="Maroon" title="c21">P29</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:63_1_1_1_1_3_1"/>
( ( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> &amp; <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> ) or ( <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:63_1_1_1_1"><span class="lab"><font color="Green" title="E56">A15</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_1_3_1_1"/>
( <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> &amp; <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> )
 ; <a class="txt" 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font></span><br/><div class="add"><a NAME="E2:63_1_1_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">P19</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c8">X2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c23">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:63"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E17:63_1_1_1_1"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></span>;<br/><a NAME="E3:63_1_1_1_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">P19</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c8">X2</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c21">P29</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1_1_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></span>;<br/><span class="kw">hence </span><a NAME="E4:63_1_1_1_1_3_1_1"/>
 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font>
 ; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_1_3_1_2"/>
( <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> )
 ; <a class="txt" 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font></span><br/><div class="add"><a NAME="E2:63_1_1_1_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">P19</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">X2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c23">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:63"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E17:63_1_1_1_1"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></span>;<br/><a NAME="E3:63_1_1_1_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">P19</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">X2</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c21">P29</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:63_1_1_1_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></span>;<br/><span class="kw">hence </span><a NAME="E4:63_1_1_1_1_3_1_2"/>
 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font>
 ; <a class="txt" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E21:63_1_1_1_1"/>
 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font>
 ; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:63_1_1_1_2"/>
 not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> st <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c13">C</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T172" title="GLIB_001:th.172">GLIB_001:172</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c16">P</font> = <font color="Maroon" title="c15">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font></span><br/><span class="kw">thus </span><a NAME="E3:63_1_1_1_2"/>
<font color="Maroon" title="c16">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="glib_001.html#T19" title="GLIB_001:th.19">GLIB_001:19</a></span>; <a class="txt" 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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:63_1"/>
 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="c4">G2</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="c9">u</font>,<font color="Maroon" title="c10">v</font>
 ; <a class="txt" 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="E12:63"/>
<font color="Maroon" title="c4">G2</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" target="_self" title="GLIB_002:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
