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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><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">G</font>;<br/>



<div><i><font color="Green" title="E88">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>now </b></a><div class="add"><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="c1">G</font>;<br/><b>assume </b><a NAME="E1:92_1"/><i><font color="Green" title="E89">A2</font></i>: 
<font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> be  <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> ;<br/><b>assume </b><a NAME="E1:92_1_1"/><i><font color="Green" title="E90">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> )
 ;<br/><a NAME="E2:92_1_1"/><b>then </b>
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:92_1_1"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:92_1_1"><i><font color="Green" title="E90">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:92_1_1"/><b>then </b><i><font color="Green" title="E91">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c3">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E45"><i><font color="Green" title="E37">Lm8</font></i></a></i>;<br/><a NAME="E5:92_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:92_1_1"><i><font color="Green" title="E90">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:92_1_1"/><b>then </b>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:92_1_1"><i><font color="Green" title="E90">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:92_1_1"/><b>then </b><i><font color="Green" title="E92">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</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"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c3">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E45"><i><font color="Green" title="E37">Lm8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">lenW</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font> 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>  ;<br/><b>reconsider </b><font color="Maroon" title="c7">rn</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">lenW</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <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>  <b>by </b><i><a class="txt" href="glib_001.html#E7:92_1_1"><i><font color="Green" title="E92">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">rm</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">lenW</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  <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>  <b>by </b><i><a class="txt" href="glib_001.html#E4:92_1_1"><i><font color="Green" title="E91">A4</font></i></a></i>;<br/><a NAME="E11:92_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">n</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> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:92_1_1"><i><font color="Green" title="E90">A3</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E12:92_1_1"/><b>then </b><i><font color="Green" title="E93">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E13:92_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">rn</font> &amp; <font color="Maroon" title="c8">rm</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> )
 <b>by </b><i><a class="txt" href="glib_001.html#E4:92_1_1"><i><font color="Green" title="E91">A4</font></i></a>, <a class="txt" href="glib_001.html#E7:92_1_1"><i><font color="Green" title="E92">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E14:92_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <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="glib_001.html#E1:92_1"><i><font color="Green" title="E89">A2</font></i></a>, <a class="txt" href="glib_001.html#E4:92_1_1"><i><font color="Green" title="E91">A4</font></i></a>, <a class="txt" href="glib_001.html#E7:92_1_1"><i><font color="Green" title="E92">A5</font></i></a>, <a class="txt" href="glib_001.html#E12:92_1_1"><i><font color="Green" title="E93">A6</font></i></a>, <a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:92_1"/>
<font color="Maroon" title="c3">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:92"/>
( <font color="Maroon" title="c2">W</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> )
 ;<br/>

<b>assume </b><a NAME="E3:92"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 ;<br/>

<a NAME="E4:92"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a> </span>)</span> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 
<b>by </b><i><a class="txt" href="glib_001.html#E1:92"><i><font color="Green" title="E88">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:92"/>
<font color="Maroon" title="c2">W</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 <b>by </b><i><a class="ref" href="finseq_6.html#T29" title="FINSEQ_6:th.29">FINSEQ_6:29</a></i>;<br/>


</div>
