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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/>



<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:97_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 ;<br/><a NAME="E2:97_1"/><b>then </b>
( <font color="Maroon">W</font> is <a href="glib_001.html#V4" target="_self">Trail-like</a> &amp; ( for <font color="Olive">m</font>, <font color="Olive">n</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> ) ) )
 <b>by </b><i/>;<br/><a NAME="E3:97_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a>  is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">lenW</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:97_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E2:97_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:97_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E4:97_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:97_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</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">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:97_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">rm</font> = <span class="p1">(<span class="default"><font color="Maroon">lenW</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E7:97_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E8:97_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:97_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</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">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:97_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">rn</font> = <span class="p1">(<span class="default"><font color="Maroon">lenW</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E11:97_1_1"/>
<font color="Maroon">lenW</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E12:97_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">rn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rm</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E13:97_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">rm</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:97_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rm</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rn</font>
 <b>by </b><i>, , <a class="txt" href="glib_001.html#E1:97_1"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E15:97_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">rn</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">rm</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E16:97_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E17:97_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E18:97_1_1"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E19:97_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E20:97_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E21:97_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 ;<br/><b>hence </b><a NAME="E22:97_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:97_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a>  is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:97"/>
( <font color="Maroon">W</font> is <a href="glib_001.html#V5" target="_self">Path-like</a> implies <font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a>  is <a href="glib_001.html#V5" target="_self">Path-like</a> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E3:97"/>
<font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a>  is <a href="glib_001.html#V5" target="_self">Path-like</a>
 ;<br/>

<a NAME="E4:97"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K6" target="_self">.reverse()</a> </span>)</span> <a href="glib_001.html#K6" target="_self">.reverse()</a>  is <a href="glib_001.html#V5" target="_self">Path-like</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:97"/>
<font color="Maroon">W</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i/>;<br/>


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