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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; not <font color="Olive" title="b1">P1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Olive" title="b2">P2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Olive" title="b2">P2</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a></span><br/><b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; not <font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green" title="E20">A3</font></i>: 
not <font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>and </b><br/><a NAME="E4:22"/><i><font color="Green" title="E21">A4</font></i>: 
not <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>and </b><br/><a NAME="E5:22"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>and </b><br/><a NAME="E6:22"/><i><font color="Green" title="E23">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> </span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a></span><br/>

<b>set </b><font color="Maroon" title="c4">P</font> = <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font>;<br/>
<a NAME="E7:22"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="glib_001.html#T31" title="GLIB_001:th.31">GLIB_001:31</a></i>;<br/>
<b>hence </b><a NAME="E8:22"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> &amp; not <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> )</span><br/>

<b>thus </b><a NAME="E9:22"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="#E2:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="#E4:22"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="#E5:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E6:22"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="helly.html#T18" target="_self" title="HELLY:th.18">Th17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a></span><br/>

<a NAME="E10:22"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<b>by </b><i><a class="txt" href="#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/>
<a NAME="E11:22"/><b>then </b>
not <font color="Maroon" title="c2">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/>
<a NAME="E12:22"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 
<b>by </b><i><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/>
<a NAME="E13:22"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P1</font>
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="glib_001.html#T30" title="GLIB_001:th.30">GLIB_001:30</a></i>;<br/>
<a NAME="E14:22"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 
<b>by </b><i><a class="txt" href="#E12:22"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E15:22"/>
not <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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