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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">P</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> ex <font color="Olive" title="b2">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Olive" title="b1">P</font> st <br/>( <font color="Olive" title="b2">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b2">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b2">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Olive" title="b1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b2">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  );<br/>
<a NAME="E1:56"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1"/>
<font color="Maroon" title="c1">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/>

<a NAME="E2:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E4:56_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T22" onmouseover="rs('glib_000/T22')" onmouseout="rh()">GLIB_000:22</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c3">P0</font> = <font color="Maroon" title="c1">P</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c2">v</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:56_1"/>
<font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="glib_001.html#T90" onmouseover="rs('glib_001/T90')" onmouseout="rh()">GLIB_001:90</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E6:56_1"/>
<font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E7:56_1"/>
( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:56"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_2"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:56_2"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/>

<a NAME="E3:56_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
;<br/>
<a NAME="E4:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A7</font></span>: 
 not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>;<br/>
<a NAME="E5:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="glib_013.html#K12" title="GLIB_013:func.12">.minDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T22" target="_self" onmouseover="rs('glpacy00/T22')" onmouseout="rh()">Th30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">v</font> =  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>;<br/>
<a NAME="E6:56_2"/>
 the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="glib_013.html#D15" onmouseover="rs('glib_013/D15')" onmouseout="rh()">GLIB_013:def 15</a></span>;<br/>
<a NAME="E7:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A9</font></span>: 
 the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">H</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">P</font>, the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">H</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">P</font>, the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> as   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_2"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="glpacy00.html#T13" target="_self" onmouseover="rs('glpacy00/T13')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E9:56_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="glib_000.html#T48" onmouseover="rs('glib_000/T48')" onmouseout="rh()">GLIB_000:48</a></span>;<br/>
<a NAME="E10:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<font color="Maroon" title="c5">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E37">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">P9</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c5">H</font><span class="kw"> such that </span><br/><a NAME="E12:56_2"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
( <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">H</font> &amp; <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">H</font> )
 <span class="kw">and </span><br/><a NAME="E13:56_2"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
(  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> iff  not <font color="Maroon" title="c5">H</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E14:56_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
( <font color="Maroon" title="c6">P9</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  iff <font color="Maroon" title="c5">H</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E15:56_2"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c6">P9</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2"><span class="lab"><font color="Green" title="E36">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">P8</font> = <font color="Maroon" title="c6">P9</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" onmouseover="rs('glib_001/T167')" onmouseout="rh()">GLIB_001:167</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">P8</font> = <font color="Maroon" title="c7">P8</font> as   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T176" onmouseover="rs('glib_001/T176')" onmouseout="rh()">GLIB_001:176</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E19:56_2"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
(  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>, the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_2"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="glib_000.html#D51" onmouseover="rs('glib_000/D51')" onmouseout="rh()">GLIB_000:def 51</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">w</font> =  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font>;<br/>
<a NAME="E20:56_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E21:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>, the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:56_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="glib_000.html#T67" onmouseover="rs('glib_000/T67')" onmouseout="rh()">GLIB_000:67</a></span>;<br/>
<a NAME="E22:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A17</font></span>: 
 the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:56_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a></span>;<br/>
<a NAME="E23:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A18</font></span>: 
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E24:56_2"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">u</font> =  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">H</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:56_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E26:56_2"/>
 the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E27:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A20</font></span>: 
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:56_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E28:56_2"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">H</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_2_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:56_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>, the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c5">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="glib_000.html#T73" onmouseover="rs('glib_000/T73')" onmouseout="rh()">GLIB_000:73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:56_2"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
( <font color="Maroon" title="c2">P</font> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c5">H</font>,<font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c9">e</font>, the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c2">P</font> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c5">H</font>, the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c11">u</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E19:56_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="glib_008.html#T38" onmouseover="rs('glib_008/T38')" onmouseout="rh()">GLIB_008:38</a></span>;<br/>
<a NAME="E30:56_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A23</font></span>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E28:56_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="glib_006.html#D12" onmouseover="rs('glib_006/D12')" onmouseout="rh()">GLIB_006:def 12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_2_2"/>
( <font color="Maroon" title="c5">H</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  or  not <font color="Maroon" title="c5">H</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_1"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
<font color="Maroon" title="c5">H</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><div class="add"><a NAME="E2:56_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A25</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T132" onmouseover="rs('zfmisc_1/T132')" onmouseout="rh()">ZFMISC_1:132</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">P0</font> = <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e</font>;<br/><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  = 
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

<br/>.= 
<font color="Maroon" title="c11">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_1"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><a NAME="E4:56_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A26</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> , the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><a NAME="E5:56_2_2_1"/>
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a></span>;<br/><a NAME="E6:56_2_2_1"/><span class="kw">then </span>
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">P0</font> = <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e</font> as   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2_2_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="ref" href="glib_001.html#T155" onmouseover="rs('glib_001/T155')" onmouseout="rh()">GLIB_001:155</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">P0</font> = <font color="Maroon" title="c13">P0</font> as   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">P0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><span class="kw">thus </span><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2_2_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="ref" href="glib_001.html#T95" onmouseover="rs('glib_001/T95')" onmouseout="rh()">GLIB_001:95</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><span class="kw">thus </span><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2_2_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="ref" href="glib_001.html#T111" onmouseover="rs('glib_001/T111')" onmouseout="rh()">GLIB_001:111</a></span>
<br/>.= 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="glib_001.html#T110" onmouseover="rs('glib_001/T110')" onmouseout="rh()">GLIB_001:110</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><span class="lab"><font color="Green" title="E58">A27</font></span>: <font color="Maroon" title="c11">u</font> = 
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_1"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 

<br/>.= 
<font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2_2_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>
;<br/><a NAME="E12:56_2_2_1"/><span class="lab"><font color="Green" title="E59">A28</font></span>: 
 the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_2_2_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><a NAME="E13:56_2_2_1"/>
<font color="Maroon" title="c5">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2_2_1"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A29</font></span>:  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:56_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E2:56_2_2_1"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="txt" href="glpacy00.html#E37"><span class="lab"><font color="Green" title="E32">Lm6</font></span></a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56_2_2_1"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E12:56_2_2_1"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="ref" href="enumset1.html#T1" onmouseover="rs('enumset1/T1')" onmouseout="rh()">ENUMSET1:1</a></span>
;<br/><a NAME="E15:56_2_2_1"/>
( <font color="Maroon" title="c6">P9</font> <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="c11">u</font> &amp; <font color="Maroon" title="c6">P9</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="c11">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_1"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E16:56_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c11">u</font>,<font color="Maroon" title="c11">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/><a NAME="E17:56_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:56_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E28:56_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glibpre1.html#T62" onmouseover="rs('glibpre1/T62')" onmouseout="rh()">GLIBPRE1:62</a>, <a class="ref" href="glibpre1.html#T63" onmouseover="rs('glibpre1/T63')" onmouseout="rh()">GLIBPRE1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E18:56_2_2_1"/>
( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:56_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E11:56_2_2_1"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="txt" href="#E12:56_2_2_1"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E14:56_2_2_1"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_2"/><span class="lab"><font color="Green" title="E55">A30</font></span>: 
 not <font color="Maroon" title="c5">H</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><div class="add"><a NAME="E2:56_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A31</font></span>: 
 <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>;<br/><a NAME="E3:56_2_2_2"/>
<font color="Maroon" title="c11">u</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_2_2_2_1"/>
 not <font color="Maroon" title="c11">u</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:56_2_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/>
<a NAME="E3:56_2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A33</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2_2_2"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="ref" href="glib_000.html#T157" onmouseover="rs('glib_000/T157')" onmouseout="rh()">GLIB_000:157</a>, <a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a></span>;<br/>
<a NAME="E4:56_2_2_2_1"/>
<font color="Maroon" title="c11">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T9" target="_self" onmouseover="rs('glpacy00/T9')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E5:56_2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A34</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2_2_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E6:56_2_2_2_1"/>
<span class="p1">(<span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font></span>)</span> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">u</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E28:56_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glpacy00.html#T6" target="_self" onmouseover="rs('glpacy00/T6')" onmouseout="rh()">Th10</a>, <a class="ref" href="glpacy00.html#T7" target="_self" onmouseover="rs('glpacy00/T7')" onmouseout="rh()">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:56_2_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2_2_2_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="ref" href="glpacy00.html#T9" target="_self" onmouseover="rs('glpacy00/T9')" onmouseout="rh()">Th17</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:56_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c5">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D8" onmouseover="rs('glib_006/D8')" onmouseout="rh()">GLIB_006:def 8</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_2_2_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_2_2_1"/><span class="lab"><font color="Green" title="E57">A35</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><div class="add"><a NAME="E2:56_2_2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> , the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">P0</font> = <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e</font>;<br/><a NAME="E3:56_2_2_2_2_1"/>
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a></span>;<br/><a NAME="E4:56_2_2_2_2_1"/><span class="kw">then </span>
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">P0</font> = <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e</font> as   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="glib_001.html#T155" onmouseover="rs('glib_001/T155')" onmouseout="rh()">GLIB_001:155</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">P0</font> = <font color="Maroon" title="c13">P0</font> as   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">P0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><span class="kw">thus </span><span class="lab"><font color="Green" title="E59">A37</font></span>: <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="glib_001.html#T95" onmouseover="rs('glib_001/T95')" onmouseout="rh()">GLIB_001:95</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><span class="kw">thus </span><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="glib_001.html#T111" onmouseover="rs('glib_001/T111')" onmouseout="rh()">GLIB_001:111</a></span>
<br/>.= 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="glib_001.html#T110" onmouseover="rs('glib_001/T110')" onmouseout="rh()">GLIB_001:110</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_2_2_2_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( (  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) &amp; ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) &amp; ( ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:56_2_2_2_2_1_3"/>
(  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_2_2_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_2_2_2_2_1_3_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">p</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font><span class="kw"> such that </span><br/><a NAME="E3:56_2_2_2_2_1_3_1"/><span class="lab"><font color="Green" title="E60">A38</font></span>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c16">p</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D8" onmouseover="rs('glib_006/D8')" onmouseout="rh()">GLIB_006:def 8</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:56_2_2_2_2_1_3_1"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="txt" href="#E3:56_2_2_2_2_1_3_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="ref" href="glib_001.html#T143" onmouseover="rs('glib_001/T143')" onmouseout="rh()">GLIB_001:143</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:56_2_2_2_2_1_3"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_1_3_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_2_2_2_2_1_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_1_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_2_2_1_3_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A39</font></span>: <font color="Maroon" title="c15">x</font> = 
<font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>
<br/>.= 
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 

;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">u9</font> = <font color="Maroon" title="c15">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">H</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">v9</font> = <font color="Maroon" title="c16">u9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:56_2_2_2_2_1_3_2_1"/><span class="lab"><font color="Green" title="E61">A40</font></span>: 
<font color="Maroon" title="c16">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:56_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E6:56_2_2_2_2_1_3_2_1"/>
<font color="Maroon" title="c16">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:56_2_2_2"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="txt" href="#E1:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E2:56_2_2_2_2_1_3_2_1"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/><a NAME="E7:56_2_2_2_2_1_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A41</font></span>: 
<font color="Maroon" title="c16">u9</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v9</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2_2_2_2_1_3_2_1"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glpacy00.html#T5" target="_self" onmouseover="rs('glpacy00/T5')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E8:56_2_2_2_2_1_3_2_1"/>
<font color="Maroon" title="c16">u9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c5">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="txt" href="#E2:56_2_2_2_2_1_3_2_1"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E9:56_2_2_2_2_1_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">v9</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_2_2_2_2_1_3_2_1"><span class="lab"><font color="Green" title="E62">A41</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/><span class="kw">hence </span><a NAME="E10:56_2_2_2_2_1_3_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_1_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_2_2_1_3_2_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font></span><br/><div class="add"><a NAME="E2:56_2_2_2_2_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A42</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><a NAME="E3:56_2_2_2_2_1_3_2_2"/>
 the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E28:56_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glib_006.html#T141" onmouseover="rs('glib_006/T141')" onmouseout="rh()">GLIB_006:141</a>, <a class="ref" href="glib_006.html#T142" onmouseover="rs('glib_006/T142')" onmouseout="rh()">GLIB_006:142</a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_2_2_2_2_1_3_2_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_1_3_2_2"><span class="lab"><font color="Green" title="E60">A42</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E10:56_2_2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A43</font></span>: 
 <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  = 
<font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>
<br/>.= 
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 

;<br/><a NAME="E12:56_2_2_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">H</font>
 ;<br/><a NAME="E13:56_2_2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A44</font></span>: 
<font color="Maroon" title="c14">P0</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="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E2:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><a NAME="E14:56_2_2_2_2_1"/>
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c11">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/><a NAME="E15:56_2_2_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:56_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E28:56_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glibpre1.html#T62" onmouseover="rs('glibpre1/T62')" onmouseout="rh()">GLIBPRE1:62</a>, <a class="ref" href="glibpre1.html#T63" onmouseover="rs('glibpre1/T63')" onmouseout="rh()">GLIBPRE1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E16:56_2_2_2_2_1"/>
( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E60">A43</font></span></a>, <a class="txt" href="#E13:56_2_2_2_2_1"><span class="lab"><font color="Green" title="E61">A44</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_2_2_2"/><span class="lab"><font color="Green" title="E57">A45</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c11">u</font> = 
<font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T22" onmouseover="rs('glib_001/T22')" onmouseout="rh()">GLIB_001:22</a></span>
;<br/><a NAME="E3:56_2_2_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A47</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> , the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:56_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">P0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e</font>;<br/><a NAME="E4:56_2_2_2_2_2"/>
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a></span>;<br/><a NAME="E5:56_2_2_2_2_2"/><span class="kw">then </span>
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/><a NAME="E6:56_2_2_2_2_2"/><span class="kw">then </span>
 not  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T92" onmouseover="rs('glib_001/T92')" onmouseout="rh()">GLIB_001:92</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">P0</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e</font> as   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="glib_001.html#T155" onmouseover="rs('glib_001/T155')" onmouseout="rh()">GLIB_001:155</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">P0</font> = <font color="Maroon" title="c13">P0</font> as   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c2">P</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">P0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><span class="kw">thus </span><span class="lab"><font color="Green" title="E59">A48</font></span>: <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="glib_001.html#T95" onmouseover="rs('glib_001/T95')" onmouseout="rh()">GLIB_001:95</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T92" onmouseover="rs('glib_001/T92')" onmouseout="rh()">GLIB_001:92</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><span class="kw">thus </span><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="glib_001.html#T111" onmouseover="rs('glib_001/T111')" onmouseout="rh()">GLIB_001:111</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T107" onmouseover="rs('glib_001/T107')" onmouseout="rh()">GLIB_001:107</a></span>
<br/>.= 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="glib_001.html#T110" onmouseover="rs('glib_001/T110')" onmouseout="rh()">GLIB_001:110</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_2_2_2_2_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( (  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) &amp; ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) &amp; ( ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:56_2_2_2_2_2_4"/>
(  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_2_2_2_2_2_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_2_2_2_2_2_4_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">p</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font><span class="kw"> such that </span><br/><a NAME="E3:56_2_2_2_2_2_4_1"/><span class="lab"><font color="Green" title="E60">A49</font></span>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font> &amp; <font color="Maroon" title="c16">p</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D8" onmouseover="rs('glib_006/D8')" onmouseout="rh()">GLIB_006:def 8</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:56_2_2_2_2_2_4_1"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E59">A48</font></span></a>, <a class="txt" href="#E3:56_2_2_2_2_2_4_1"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="ref" href="glib_001.html#T143" onmouseover="rs('glib_001/T143')" onmouseout="rh()">GLIB_001:143</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:56_2_2_2_2_2_4"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_2_4_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_2_2_2_2_2_4_2"/><span class="kw">then </span>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_2_4_2_1"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_2_2_2_4_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A50</font></span>: <font color="Maroon" title="c15">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>
<br/>.= 
<font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T22" onmouseover="rs('glib_001/T22')" onmouseout="rh()">GLIB_001:22</a></span>
<br/>.= 
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">u9</font> = <font color="Maroon" title="c15">x</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">H</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">v9</font> = <font color="Maroon" title="c16">u9</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:56_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:56_2_2_2_2_2_4_2_1"/><span class="lab"><font color="Green" title="E61">A51</font></span>: 
<font color="Maroon" title="c16">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:56_2"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/><a NAME="E6:56_2_2_2_2_2_4_2_1"/>
<font color="Maroon" title="c16">u9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c9">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:56_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:56_2_2_2"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="txt" href="#E1:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="txt" href="#E2:56_2_2_2_2_2_4_2_1"><span class="lab"><font color="Green" title="E60">A50</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/><a NAME="E7:56_2_2_2_2_2_4_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A52</font></span>: 
<font color="Maroon" title="c16">u9</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v9</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2_2_2_2_2_4_2_1"><span class="lab"><font color="Green" title="E61">A51</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glpacy00.html#T5" target="_self" onmouseover="rs('glpacy00/T5')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E8:56_2_2_2_2_2_4_2_1"/>
<font color="Maroon" title="c16">u9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c5">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2"><span class="lab"><font color="Green" title="E56">A31</font></span></a>, <a class="txt" href="#E2:56_2_2_2_2_2_4_2_1"><span class="lab"><font color="Green" title="E60">A50</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E9:56_2_2_2_2_2_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">v9</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_2_2_2_2_2_4_2_1"><span class="lab"><font color="Green" title="E62">A52</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/><span class="kw">hence </span><a NAME="E10:56_2_2_2_2_2_4_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_2_2_2_2_4_2_2"><span class="kw">suppose </span></a><a NAME="E1:56_2_2_2_2_2_4_2_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font></span><br/><div class="add"><a NAME="E2:56_2_2_2_2_2_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A53</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><a NAME="E3:56_2_2_2_2_2_4_2_2"/>
 the   <a href="glib_013.html#V9" title="GLIB_013:attr.9">with_min_degree</a>  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E28:56_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glib_006.html#T141" onmouseover="rs('glib_006/T141')" onmouseout="rh()">GLIB_006:141</a>, <a class="ref" href="glib_006.html#T142" onmouseover="rs('glib_006/T142')" onmouseout="rh()">GLIB_006:142</a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_2_2_2_2_2_4_2_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2_2_2_2_2_4_2_2"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E12:56_2_2_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A54</font></span>: 
 <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>
<br/>.= 
<font color="Maroon" title="c8">P8</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T22" onmouseover="rs('glib_001/T22')" onmouseout="rh()">GLIB_001:22</a></span>
<br/>.= 
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

;<br/><a NAME="E14:56_2_2_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">H</font>
 ;<br/><a NAME="E15:56_2_2_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A55</font></span>: 
<font color="Maroon" title="c14">P0</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="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:56_2"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E3:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E58">A47</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><a NAME="E16:56_2_2_2_2_2"/>
<font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a>  <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c11">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a>, <a class="ref" href="glib_001.html#T23" onmouseover="rs('glib_001/T23')" onmouseout="rh()">GLIB_001:23</a></span>;<br/><a NAME="E17:56_2_2_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:56_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E28:56_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E29:56_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="glibpre1.html#T62" onmouseover="rs('glibpre1/T62')" onmouseout="rh()">GLIBPRE1:62</a>, <a class="ref" href="glibpre1.html#T63" onmouseover="rs('glibpre1/T63')" onmouseout="rh()">GLIBPRE1:63</a></span>;<br/><span class="kw">hence </span><a NAME="E18:56_2_2_2_2_2"/>
( (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c2">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c14">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E60">A54</font></span></a>, <a class="txt" href="#E15:56_2_2_2_2_2"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:56"/><span class="lab"><font color="Green" title="E36">A58</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E35">A4</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">P</font> st <br/>( <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Olive" title="b1">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() ) &amp; ( <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Olive" title="b1">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:56"/><span class="lab"><font color="Green" title="E37">A59</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">P0</font> being   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E7:56"/><span class="lab"><font color="Green" title="E38">A60</font></span>: 
( <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) )
 <span class="kw">and </span><br/><a NAME="E8:56"/><span class="lab"><font color="Green" title="E39">A61</font></span>: 
( ( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E36">A58</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E37">A59</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">P0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() ) &amp; ( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:56_3"/>
( <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="56_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_3_1"/>
<font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font><span class="kw"> such that </span><br/><a NAME="E3:56_3_1"/><span class="lab"><font color="Green" title="E40">A62</font></span>: 
<font color="Maroon" title="c3">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T128" onmouseover="rs('glib_001/T128')" onmouseout="rh()">GLIB_001:128</a></span>;<br/>
<a NAME="E4:56_3_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E38">A60</font></span></a>, <a class="txt" href="#E3:56_3_1"><span class="lab"><font color="Green" title="E40">A62</font></span></a>, <a class="ref" href="glib_001.html#T90" onmouseover="rs('glib_001/T90')" onmouseout="rh()">GLIB_001:90</a></span>;<br/>
<a NAME="E5:56_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">P</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T27" onmouseover="rs('glib_000/T27')" onmouseout="rh()">GLIB_000:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:56_3_1"/>
<font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:56_3"/>
<font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()</span><br/><span class="kw">hence </span><a NAME="E3:56_3"/>
<font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:56"/>
( <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P</font> &amp; (  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; (  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() ) &amp; ( <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) &amp; ( <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  ) &amp; <font color="Maroon" title="c3">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56"><span class="lab"><font color="Green" title="E38">A60</font></span></a>, <a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E39">A61</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
