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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#M2" title="GLIB_000:mode.2">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/><b>let </b><font color="Maroon" title="c2">G2</font> be    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>





<b>reconsider </b><font color="Maroon" title="c4">W'</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> <b>by </b><i><a class="ref" href="glib_001.html#T168" target="_self" title="GLIB_001:th.168">Th168</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
<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> 
 <b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self" title="GLIB_001:def.24">Def24</a></i>;<br/><a NAME="E3:246_1"/><b>then </b>
<font color="Maroon" title="c4">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="c4">W'</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><b>hence </b><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>
 <b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self" title="GLIB_001:def.24">Def24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>assume </b><a NAME="E1:246_2"/><i><font color="Green" title="E153">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">W'</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">W'</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">W'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )</span><br/><b>assume </b><a NAME="E1:246_2_1"/><i><font color="Green" title="E154">A2</font></i>: 
<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">W'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">W'</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">W'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span><br/><a NAME="E2:246_2_1"/><b>then </b><i><font color="Green" title="E155">A3</font></i>: 
<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">W'</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">W'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="#E1:246_2"><i><font color="Green" title="E153">A1</font></i></a>, <a class="ref" href="glib_001.html#D25" target="_self" title="GLIB_001:def.25">Def25</a></i>;<br/><a NAME="E3:246_2_1"/><i><font color="Green" title="E156">A4</font></i>: 
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
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E4:246_2_1"/>
<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">W'</font>
 <b>by </b><i><a class="txt" href="#E1:246_2_1"><i><font color="Green" title="E154">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:246_2_1"/><b>then </b>
<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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">W'</font>
 <b>by </b><i><a class="txt" href="#E3:246_2_1"><i><font color="Green" title="E156">A4</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:246_2_1"/><b>then </b>
<font color="Maroon" title="c4">W'</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#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c2">G2</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T9" target="_self" title="GLIB_001:th.9">Th9</a></i>;<br/><b>hence </b><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">W'</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">W'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="#E2:246_2_1"><i><font color="Green" title="E155">A3</font></i></a>, <a class="ref" href="glib_000.html#D34" title="GLIB_000:def.34">GLIB_000:def 34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><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>
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self" title="GLIB_001:def.25">Def25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E92"><i><font color="Green" title="E84">Lm55</font></i></a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="txt" href="glib_001.html#E91"><i><font color="Green" title="E83">Lm54</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><i><font color="Green" title="E153">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="246_4"><b>now </b></a><div class="add"><b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
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">W'</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">W'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><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:()"><i><font color="Red">::  thesis: </font></i></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"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:246_5"/>
<font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
( <font color="Maroon" title="c4">W'</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> &amp; ( 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">W'</font> &amp; <font color="Maroon" title="c4">W'</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">W'</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">W'</font> ) ) )
 <b>by </b><i><a class="txt" href="#E5:246"><i><font color="Green" title="E153">A5</font></i></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><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:()"><i><font color="Red">::  thesis: </font></i></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><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span>
<div class="add"><b>assume </b><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:()"><i><font color="Red">::  thesis: </font></i></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"/><b>then </b>
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">W'</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">W'</font> &amp; <font color="Maroon" title="c4">W'</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">W'</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>
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self" title="GLIB_001:def.29">Def29</a></i>;<br/><b>hence </b><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>
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self" title="GLIB_001:def.29">Def29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:246"/>
verum
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
