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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c1">G</font><br/> for <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Olive" title="b3">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; not <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; ( <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  or  not <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) holds <br/><font color="Olive" title="b1">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Olive" title="b2">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; not <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; ( <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  or  not <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) holds <br/><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b1">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">e</font>, <font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; not <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; ( <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  or  not <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) implies <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:102"/><span class="lab"><font color="Green" title="E98">A1</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E2:102"/><span class="lab"><font color="Green" title="E99">A2</font></span>: 
<font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">and </span><br/><a NAME="E3:102"/><span class="lab"><font color="Green" title="E100">A3</font></span>: 
not <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">and </span><br/><a NAME="E4:102"/><span class="lab"><font color="Green" title="E101">A4</font></span>: 
( <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  or  not <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/>

<a NAME="E5:102"/><span class="lab"><font color="Green" title="E102">A5</font></span>: 
for <font color="Olive" title="b1">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 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">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="c2">W</font> holds <br/><font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:102"><span class="lab"><font color="Green" title="E100">A3</font></span></a>, <a class="txt" href="glib_001.html#E82"><span class="lab"><font color="Green" title="E74">Lm45</font></span></a></span>;<br/>
<a NAME="E6:102"/>
not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E98">A1</font></span></a>, <a class="txt" href="#E3:102"><span class="lab"><font color="Green" title="E100">A3</font></span></a>, <a class="txt" href="glib_001.html#E85"><span class="lab"><font color="Green" title="E77">Lm48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:102"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:102"><span class="lab"><font color="Green" title="E98">A1</font></span></a>, <a class="txt" href="#E2:102"><span class="lab"><font color="Green" title="E99">A2</font></span></a>, <a class="txt" href="#E4:102"><span class="lab"><font color="Green" title="E101">A4</font></span></a>, <a class="txt" href="#E5:102"><span class="lab"><font color="Green" title="E102">A5</font></span></a>, <a class="txt" href="glib_001.html#E102"><span class="lab"><font color="Green" title="E94">Lm65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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