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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:103"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_1"><b>suppose </b></a><a NAME="E1:103_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:103_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V3" target="_self">trivial</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E97"><i><font color="Green">Lemma89</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:103_1_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E98"><i><font color="Green">Lemma90</font></i></a></i>;<br/><b>thus </b><a NAME="E5:103_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:103_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1_2"><b>suppose </b></a><a NAME="E1:103_1_1_2"/><i><font color="Green">E105</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:103_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E3:103_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:103_1_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:103_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:103_1_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="txt" href="glib_001.html#E1:103"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:103_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:103"><i><font color="Green">E104</font></i></a>, <a class="txt" href="glib_001.html#E103"><i><font color="Green">Lemma95</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:103_1_1_2_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E2:103_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E5:103_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_001.html#E1:103_1_1_2_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:103_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 3
 <b>by </b><i><a class="ref" href="glib_001.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:103_1_1_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="xcmplx_0.html#K6">-</a> 2
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E5:103_1_1_2_1"/><i><font color="Green">E110</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>hence </b><a NAME="E6:103_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E4:103_1_1_2_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:103_1_1_2_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:103_1_1_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_001.html#E5:103_1_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:103_1_1_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="glib_001.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E9:103_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:103_1_1_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_001.html#E1:103_1_1_2_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:103_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:103_1_1_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="glib_001.html#D28" target="_self">Def28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:103"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 ;<br/>


</div>
