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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font> be   <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">G2</font> being    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Olive" title="b1">G2</font> holds <br/> ( ( <font color="Olive" title="b2">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Olive" title="b2">W</font> is   <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b2">W</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  implies <font color="Olive" title="b2">W</font> is   <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b2">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  implies <font color="Olive" title="b2">W</font> is   <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b2">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>  implies <font color="Olive" title="b2">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b2">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Olive" title="b2">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b2">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Olive" title="b2">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font> holds <br/> ( ( <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  implies <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  implies <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>  implies <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Olive" title="b1">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</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="c3">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">W9</font> = <font color="Maroon" title="c3">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" target="_self" title="GLIB_001:th.167">Th168</a></span>;<br/>
<div><span class="kw">hereby </span> <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="c3">W</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:246_1"/>
<font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</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="c3">W</font> is   <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font></span><br/><a NAME="E2:246_1"/><span class="kw">then </span>
<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="ref" href="glib_001.html#D24" target="_self" title="GLIB_001:def.24">Def24</a></span>;<br/><a NAME="E3:246_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">W9</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="c4">W9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><span class="kw">hence </span><a NAME="E4:246_1"/>
<font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D24" target="_self" title="GLIB_001:def.24">Def24</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">hereby </span> <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="c3">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:246_2"/><span class="lab"><font color="Green" title="E153">A1</font></span>: 
<font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</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="c3">W</font> is   <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="246_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <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> ; <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="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W9</font> implies <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )</span><br/><a NAME="E1:246_2_1"/><span class="lab"><font color="Green" title="E154">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</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">assume </span><a NAME="E2:246_2_1"/><span class="lab"><font color="Green" title="E155">A3</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span><br/><a NAME="E3:246_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</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> <font color="Maroon" title="c4">W9</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/><a NAME="E4:246_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">W9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:246_2_1"><span class="lab"><font color="Green" title="E154">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:246_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A4</font></span>: 
<font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T8" target="_self" title="GLIB_001:th.8">Th9</a></span>;<br/><a NAME="E6:246_2_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:246_2"><span class="lab"><font color="Green" title="E153">A1</font></span></a>, <a class="txt" href="#E2:246_2_1"><span class="lab"><font color="Green" title="E155">A3</font></span></a>, <a class="ref" href="glib_001.html#D25" target="_self" title="GLIB_001:def.25">Def25</a></span>;<br/><span class="kw">hence </span><a NAME="E7:246_2_1"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:246_2_1"><span class="lab"><font color="Green" title="E156">A4</font></span></a>, <a class="ref" href="glib_000.html#D32" title="GLIB_000:def.32">GLIB_000:def 32</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">hence </span><a NAME="E3:246_2"/>
<font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D25" target="_self" title="GLIB_001:def.25">Def25</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">hereby </span> <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="c3">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:246_3"/>
<font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</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="c3">W</font> is   <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font></span><br/><a NAME="E2:246_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E92"><span class="lab"><font color="Green" title="E84">Lm55</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:246_3"/>
<font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E91"><span class="lab"><font color="Green" title="E83">Lm54</font></span></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="lab"><font color="Green" title="E153">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="246_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:246_4"/>
<font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</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="c3">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font></span><br/><a NAME="E2:246_4"/><span class="kw">then </span>
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</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> <font color="Maroon" title="c3">W</font> holds <br/><font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:246_4"/>
<font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></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">hence </span><a NAME="E6:246"/>
( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>   <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"> ( ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="246_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:246_5"/><span class="lab"><font color="Green" title="E154">A6</font></span>: 
<font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</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="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font></span><br/><a NAME="E2:246_5"/><span class="kw">then </span>
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</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> <font color="Maroon" title="c4">W9</font> &amp; <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> holds <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:246_5"/>
<font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:246"><span class="lab"><font color="Green" title="E153">A5</font></span></a>, <a class="txt" href="#E1:246_5"><span class="lab"><font color="Green" title="E154">A6</font></span></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</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">hence </span><a NAME="E8:246"/>
( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <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"> ( <font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  implies <font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:246_6"/>
<font color="Maroon" title="c3">W</font> is  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</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="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font></span><br/><a NAME="E2:246_6"/><span class="kw">then </span>
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</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>   st <font color="Olive" title="b1">m</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> <font color="Maroon" title="c4">W9</font> &amp; <font color="Olive" title="b2">n</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> <font color="Maroon" title="c4">W9</font> &amp; <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D29" target="_self" title="GLIB_001:def.29">Def29</a></span>;<br/><span class="kw">hence </span><a NAME="E3:246_6"/>
<font color="Maroon" title="c3">W</font> is   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D29" target="_self" title="GLIB_001:def.29">Def29</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>
