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<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_000.html#M2">Subgraph</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:247"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:247"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V1" target="_self">closed</a> iff <font color="Maroon">c<sub>3</sub></font> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="glib_001.html#K4" target="_self">.last()</a>  )
 
<b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self">Def24</a></i>;<br/>
<a NAME="E3:247"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#K3" target="_self">.first()</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#K4" target="_self">.last()</a>  )
 
<b>by </b><i><a class="txt" href="glib_001.html#E1:247"><i><font color="Green">E161</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V1" target="_self">closed</a> iff <font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V1" target="_self">closed</a> )
 <b>by </b><i><a class="txt" href="glib_001.html#E2:247"><i><font color="Green">E162</font></i></a>, <a class="ref" href="glib_001.html#D24" target="_self">Def24</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="247_1"><b>now </b></a><div class="add"><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:247_1_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V2" target="_self">directed</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="247_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:247_1_1_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:247_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:247_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:247_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="glib_001.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E5:247_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D34">GLIB_000:def 34</a></i>;<br/><b>hence </b><a NAME="E6:247_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:247"><i><font color="Green">E161</font></i></a>, <a class="txt" href="glib_001.html#E1:247_1_1"><i><font color="Green">E163</font></i></a>, <a class="txt" href="glib_001.html#E1:247_1_1_1"><i><font color="Green">E164</font></i></a>, <a class="ref" href="glib_001.html#D25" target="_self">Def25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:247_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V2" target="_self">directed</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:247_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V2" target="_self">directed</a>
 ;<br/><b>hence </b><a NAME="E3:247_1"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="glib_001.html#V2" target="_self">directed</a>  <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:247"><i><font color="Green">E161</font></i></a>, <a class="ref" href="glib_001.html#T176" target="_self">Th176</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V2" target="_self">directed</a> iff <font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V2" target="_self">directed</a> )
 ;<br/>

<a NAME="E7:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V3" target="_self">trivial</a> iff  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="glib_001.html#E1:247"><i><font color="Green">E161</font></i></a>, <a class="txt" href="glib_001.html#E97"><i><font color="Green">Lemma89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V3" target="_self">trivial</a> iff <font color="Maroon">c<sub>4</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/>

<a NAME="E9:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a> iff for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="glib_001.html#E1:247"><i><font color="Green">E161</font></i></a>, <a class="txt" href="glib_001.html#E99"><i><font color="Green">Lemma91</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:247"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a> iff <font color="Maroon">c<sub>4</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#E99"><i><font color="Green">Lemma91</font></i></a></i>;<br/>

<a NAME="E11:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V5" target="_self">Path-like</a> iff ( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> ) ) ) )
 
<b>by </b><i><a class="txt" href="glib_001.html#E1:247"><i><font color="Green">E161</font></i></a>, <a class="ref" href="glib_001.html#D28" target="_self">Def28</a></i>;<br/>
<b>hence </b><a NAME="E12:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V5" target="_self">Path-like</a> iff <font color="Maroon">c<sub>4</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#E10:247"><i><font color="Green">E163</font></i></a>, <a class="ref" href="glib_001.html#D28" target="_self">Def28</a></i>;<br/>

<a NAME="E13:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V6" target="_self">vertex-distinct</a> iff for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="glib_001.html#E1:247"><i><font color="Green">E161</font></i></a>, <a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>;<br/>
<b>hence </b><a NAME="E14:247"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="glib_001.html#V6" target="_self">vertex-distinct</a> iff <font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V6" target="_self">vertex-distinct</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>;<br/>


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