<?xml version="1.0"?>
<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>   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</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">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b4">C</font> being    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">v2</font> &amp;  not <font color="Olive" title="b3">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="c1">P</font> holds <br/><font color="Olive" title="b4">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b2">C</font> being    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">v1</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">v2</font>  st <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> &amp;  not <font color="Olive" title="b1">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="c1">P</font> holds <br/><font color="Olive" title="b2">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">e</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">  for <font color="Olive" title="b1">C</font> being    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">v2</font>  st <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> &amp;  not <font color="Maroon" title="c4">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="c1">P</font> holds <br/><font color="Olive" title="b1">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">C</font> be    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c3">v2</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="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> &amp;  not <font color="Maroon" title="c4">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="c1">P</font> implies <font color="Maroon" title="c5">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:92"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> &amp;  not <font color="Maroon" title="c4">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="c1">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="c5">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> </span><br/>

<a NAME="E2:92"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T29" target="_self" onmouseover="rs('glpacy00/T29')" onmouseout="rh()">Th37</a></span>;<br/>
<a NAME="E3:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c7">v2</font> = <font color="Maroon" title="c3">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E5:92"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c5">C</font> is    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c7">v2</font>
 
;<br/>
<a NAME="E6:92"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
( <font color="Maroon" title="c6">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp; <font color="Maroon" title="c7">v2</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="#E3:92"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_1"><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">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">C</font> holds  <font color="Olive" title="b1">v</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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">C</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="Olive">b<sub>1</sub></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><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">w</font> = <font color="Maroon" title="c8">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="glib_006.html#T102" onmouseover="rs('glib_006/T102')" onmouseout="rh()">GLIB_006:102</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1"><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="E1:92_1_1"/>
( ( <font color="Maroon" title="c9">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c9">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> ) or <font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_1"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
( <font color="Maroon" title="c9">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c9">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</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="Olive">b<sub>1</sub></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><br/><div class="add"><a NAME="E2:92_1_1_1"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
 not <font color="Maroon" title="c9">w</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="92_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:92_1_1_1_1"/>
<font color="Maroon" title="c9">w</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="E2:92_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">w</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="c1">P</font> &amp; <font color="Maroon" title="c6">v1</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="c1">P</font> &amp; <font color="Maroon" title="c7">v2</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="c1">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="glib_006.html#D8" onmouseover="rs('glib_006/D8')" onmouseout="rh()">GLIB_006:def 8</a></span>;<br/>
<a NAME="E3:92_1_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_2.html#T1" onmouseover="rs('nomin_2/T1')" onmouseout="rh()">NOMIN_2:1</a>, <a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_1:11</a></span>;<br/>
<a NAME="E4:92_1_1_1_1"/><span class="kw">then </span>
3 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E1:92_1_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="card_2.html#T58" onmouseover="rs('card_2/T58')" onmouseout="rh()">CARD_2:58</a></span>;<br/>
<a NAME="E5:92_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T29" target="_self" onmouseover="rs('glpacy00/T29')" onmouseout="rh()">Th37</a>, <a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a></span>;<br/>
<a NAME="E6:92_1_1_1_1"/>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E7:92_1_1_1_1"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:92_1_1_1_1"><span class="lab"><font color="Green" title="E54">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:92_1_1_1_1"/>
contradiction
 ; <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">thus </span><font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  = 
<font color="Maroon" title="c9">w</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="#E1:92_1_1_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="glibpre0.html#T46" onmouseover="rs('glibpre0/T46')" onmouseout="rh()">GLIBPRE0:46</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_1_1_1"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="glpacy00.html#T27" target="_self" onmouseover="rs('glpacy00/T27')" onmouseout="rh()">Th35</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="92_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_2"/>
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</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="Olive">b<sub>1</sub></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><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</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="#E1:92"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="glibpre0.html#T47" onmouseover="rs('glibpre0/T47')" onmouseout="rh()">GLIBPRE0:47</a></span>
<br/>.= 
1 <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="#E6:92"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>
<br/>.= 
2

;<br/> <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="92_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:92_1_1_3"/>
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</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="Olive">b<sub>1</sub></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><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v2</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="#E1:92"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E5:92"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="glibpre0.html#T48" onmouseover="rs('glibpre0/T48')" onmouseout="rh()">GLIBPRE0:48</a></span>
<br/>.= 
1 <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="#E6:92"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>
<br/>.= 
2

;<br/> <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="E8:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c5">C</font> is 2 <a href="glib_016.html#V2" title="GLIB_016:attr.2">-regular</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_016.html#D4" onmouseover="rs('glib_016/D4')" onmouseout="rh()">GLIB_016:def 4</a></span>;<br/>
<a NAME="E9:92"/><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c5">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="glib_006.html#T105" onmouseover="rs('glib_006/T105')" onmouseout="rh()">GLIB_006:105</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_2"><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">  ex <font color="Olive" title="b1">C0</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">C</font> st <font color="Olive" title="b1">C0</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">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="E2:92_2"/>
( <font color="Maroon" title="c8">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="c8">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">and </span><br/><a NAME="E3:92_2"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
(  <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="c8">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="c8">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> iff  not <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E4:92_2"/><span class="lab"><font color="Green" title="E55">A12</font></span>: 
( <font color="Maroon" title="c8">P0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>() iff <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E5:92_2"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
( <font color="Maroon" title="c8">P0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>  iff <font color="Maroon" title="c1">P</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">and </span><br/><a NAME="E6:92_2"/>
<font color="Maroon" title="c8">P0</font> is  <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T23" target="_self" onmouseover="rs('glpacy00/T23')" onmouseout="rh()">Th31</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">P9</font> = <font color="Maroon" title="c8">P0</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T75" onmouseover="rs('glib_006/T75')" onmouseout="rh()">GLIB_006:75</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c10">C0</font> = <font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</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="Olive">b<sub>1</sub></font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> </span><br/><a NAME="E8:92_2"/><span class="lab"><font color="Green" title="E57">A14</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c5">C</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:92_2_1"/>
( ( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) &amp; ( <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="#E3:92_2"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_1_1"><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="E1:92_2_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  ) or ( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</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="#E1:92"><span class="lab"><font color="Green" title="E47">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_2_1_1_1"/>
( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c5">C</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_1_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</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="92_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_2_1_1_2"/>
( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c7">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c5">C</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_1_1_2"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</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="E9:92_2"/>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:92_2"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="ref" href="glib_001.html#T125" onmouseover="rs('glib_001/T125')" onmouseout="rh()">GLIB_001:125</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_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="E1:92_2_2"/><span class="kw">then </span>
( 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P9</font> or 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P9</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_1"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_1"/><span class="lab"><font color="Green" title="E58">A15</font></span>: 
3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P9</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="Olive">b<sub>1</sub></font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> </span><br/><div class="add"><a NAME="E2:92_2_2_1"/>
<font color="Maroon" title="c1">P</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c5">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/><a NAME="E3:92_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A16</font></span>: 
<font color="Maroon" title="c9">P9</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T176" onmouseover="rs('glib_001/T176')" onmouseout="rh()">GLIB_001:176</a></span>;<br/><a NAME="E4:92_2_2_1"/>
<font color="Maroon" title="c9">P9</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:92_2"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="ref" href="glib_001.html#T121" onmouseover="rs('glib_001/T121')" onmouseout="rh()">GLIB_001:121</a></span>;<br/><span class="kw">hence </span><a NAME="E5:92_2_2_1"/>
<font color="Maroon" title="c10">C0</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E1:92_2_2_1"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="txt" href="#E3:92_2_2_1"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="ref" href="chord.html#T33" onmouseover="rs('chord/T33')" onmouseout="rh()">CHORD:33</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="92_2_2_2"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2"/><span class="lab"><font color="Green" title="E58">A17</font></span>: 
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P9</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="Olive">b<sub>1</sub></font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:92_2_2_2"/><span class="lab"><font color="Green" title="E59">A18</font></span>: 
<font color="Maroon" title="c11">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E4:92_2_2_2"/><span class="lab"><font color="Green" title="E60">A19</font></span>: 
<font color="Maroon" title="c8">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<font color="Maroon" title="c11">e0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T28" onmouseover="rs('glibpre1/T28')" onmouseout="rh()">GLIBPRE1:28</a></span>;<br/><font color="Maroon" title="c10">C0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  = 
<font color="Maroon" title="c9">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="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</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="c10">C0</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="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</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="E6:92_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A21</font></span>: 
<font color="Maroon" title="c10">C0</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/><span class="lab"><font color="Green" title="E62">A22</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P9</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="glib_001.html#T64" onmouseover="rs('glib_001/T64')" onmouseout="rh()">GLIB_001:64</a></span>
<br/>.= 
5
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_2_2_2"><span class="lab"><font color="Green" title="E58">A17</font></span></a></span>
;<br/><a NAME="E8:92_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A23</font></span>: 
<font color="Maroon" title="c10">C0</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" onmouseover="rs('glib_001/T126')" onmouseout="rh()">GLIB_001:126</a></span>;<br/><a NAME="E9:92_2_2_2"/><span class="lab"><font color="Green" title="E64">A24</font></span>: 
<font color="Maroon" title="c10">C0</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_2_2_2_3"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E65">A25</font></span>: <font color="Maroon" title="c10">C0</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">P9</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="glib_001.html#T82" onmouseover="rs('glib_001/T82')" onmouseout="rh()">GLIB_001:82</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T86" onmouseover="rs('glib_001/T86')" onmouseout="rh()">GLIB_001:86</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c11">e0</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92_2_2_2"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E4:92_2_2_2"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="glib_001.html#T83" onmouseover="rs('glib_001/T83')" onmouseout="rh()">GLIB_001:83</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c11">e0</font>,<font color="Maroon" title="c4">e</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D9" onmouseover="rs('finseq_1/D9')" onmouseout="rh()">FINSEQ_1:def 9</a></span>
;<br/>
<a NAME="E2:92_2_2_2_3"/>
<font color="Maroon" title="c11">e0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E3:92_2_2_2"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<a NAME="E3:92_2_2_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">C0</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_2_2_2_3"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="ref" href="finseq_3.html#T94" onmouseover="rs('finseq_3/T94')" onmouseout="rh()">FINSEQ_3:94</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:92_2_2_2_3"/>
<font color="Maroon" title="c10">C0</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D27" onmouseover="rs('glib_001/D27')" onmouseout="rh()">GLIB_001:def 27</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="E10:92_2_2_2"/>
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> &amp; <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> holds <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_2_2_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c13">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> &amp; <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> implies ( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> ) )</span><br/>

<a NAME="E1:92_2_2_2_4"/><span class="lab"><font color="Green" title="E65">A26</font></span>: 
( 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">P9</font> &amp; 3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">P9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_2_2_2"><span class="lab"><font color="Green" title="E58">A17</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E2:92_2_2_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A27</font></span>: 
<font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="glib_001.html#T65" onmouseover="rs('glib_001/T65')" onmouseout="rh()">GLIB_001:65</a></span>;<br/>
<span class="lab"><font color="Green" title="E67">A28</font></span>: <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<font color="Maroon" title="c9">P9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E1:92_2_2_2_4"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="ref" href="glib_001.html#T65" onmouseover="rs('glib_001/T65')" onmouseout="rh()">GLIB_001:65</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<font color="Maroon" title="c11">e0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92_2_2_2"><span class="lab"><font color="Green" title="E59">A18</font></span></a>, <a class="txt" href="#E4:92_2_2_2"><span class="lab"><font color="Green" title="E60">A19</font></span></a>, <a class="ref" href="glib_001.html#D5" onmouseover="rs('glib_001/D5')" onmouseout="rh()">GLIB_001:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

;<br/>
<span class="lab"><font color="Green" title="E68">A29</font></span>: <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 = 
<font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">P9</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:92_2_2_2"><span class="lab"><font color="Green" title="E58">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">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="#E8:92_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="glib_001.html#T65" onmouseover="rs('glib_001/T65')" onmouseout="rh()">GLIB_001:65</a></span>
;<br/>
<a NAME="E5:92_2_2_2_4"/><span class="lab"><font color="Green" title="E69">A30</font></span>: 
<font color="Maroon" title="c8">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="c8">P0</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="92_2_2_2_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:92_2_2_2_4_3"/>
<font color="Maroon" title="c8">P0</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="c8">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"> contradiction</span><br/>

<a NAME="E2:92_2_2_2_4_3"/><span class="kw">then </span>
 <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#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P0</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span></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="#E3:92_2"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:92_2_2_2_4_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</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="E6:92_2_2_2_4"/><span class="lab"><font color="Green" title="E70">A31</font></span>: 
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> &amp; <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">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" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/>

<a NAME="E7:92_2_2_2_4"/><span class="kw">then </span>
 not  not <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:92_2_2_2"><span class="lab"><font color="Green" title="E62">A22</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_2_4_4"><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="E1:92_2_2_2_4_4"/><span class="kw">then </span>
( <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_2_4_4_1"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_1"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_2_2_4_4_1"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_2_4_4_2"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_2"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.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"> ( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_2_2_4_4_2"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92_2_2_2_4"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="ref" href="chord.html#T2" onmouseover="rs('chord/T2')" onmouseout="rh()">CHORD:2</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="92_2_2_2_4_4_3"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_3"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_2_2_4_4_3"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T5" onmouseover="rs('polyform/T5')" onmouseout="rh()">POLYFORM:5</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="92_2_2_2_4_4_4"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_4"/><span class="lab"><font color="Green" title="E71">A33</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <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="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><a NAME="E2:92_2_2_2_4_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A34</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92_2_2_2_4"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="ref" href="chord.html#T3" onmouseover="rs('chord/T3')" onmouseout="rh()">CHORD:3</a></span>;<br/><a NAME="E3:92_2_2_2_4_4_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" onmouseover="rs('chord/T2')" onmouseout="rh()">CHORD:2</a></span>;<br/><a NAME="E4:92_2_2_2_4_4_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_2_2_2_4_4_4"><span class="lab"><font color="Green" title="E72">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:92_2_2_2_4_4_4"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_2_2_2_4"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="txt" href="#E3:92_2_2_2_4"><span class="lab"><font color="Green" title="E67">A28</font></span></a>, <a class="txt" href="#E5:92_2_2_2_4"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="#E6:92_2_2_2_4"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="txt" href="#E1:92_2_2_2_4_4_4"><span class="lab"><font color="Green" title="E71">A33</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_2_4_4_5"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_5"/>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <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="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_2_2_4_4_5"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T7" onmouseover="rs('polyform/T7')" onmouseout="rh()">POLYFORM:7</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="92_2_2_2_4_4_6"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_6"/><span class="lab"><font color="Green" title="E71">A36</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <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="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><a NAME="E2:92_2_2_2_4_4_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A37</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92_2_2_2_4"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="ref" href="chord.html#T3" onmouseover="rs('chord/T3')" onmouseout="rh()">CHORD:3</a></span>;<br/><a NAME="E3:92_2_2_2_4_4_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" onmouseover="rs('chord/T2')" onmouseout="rh()">CHORD:2</a></span>;<br/><a NAME="E4:92_2_2_2_4_4_6"/><span class="kw">then </span>
 not  not <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp;  ...  &amp;  not <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92_2_2_2_4_4_6"><span class="lab"><font color="Green" title="E72">A37</font></span></a>, <a class="ref" href="nat_1.html#T62" onmouseover="rs('nat_1/T62')" onmouseout="rh()">NAT_1:62</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_2_4_4_6_1"><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="E1:92_2_2_2_4_4_6_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_2_4_4_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_6_1_1"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.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"> ( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_2_2_4_4_6_1_1"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:92_2_2_2"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="txt" href="#E1:92_2_2_2_4_4_6"><span class="lab"><font color="Green" title="E71">A36</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_2_2_2_4_4_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_6_1_2"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_2_2_4_4_6_1_2"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T5" onmouseover="rs('polyform/T5')" onmouseout="rh()">POLYFORM:5</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="92_2_2_2_4_4_6_1_3"><span class="kw">suppose </span></a><a NAME="E1:92_2_2_2_4_4_6_1_3"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <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="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:92_2_2_2_4_4_6_1_3"/>
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">C0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:92_2_2_2_4"><span class="lab"><font color="Green" title="E67">A28</font></span></a>, <a class="txt" href="#E4:92_2_2_2_4"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="txt" href="#E5:92_2_2_2_4"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="txt" href="#E6:92_2_2_2_4"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="txt" href="#E1:92_2_2_2_4_4_6"><span class="lab"><font color="Green" title="E71">A36</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></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="E11:92_2_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">C0</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:92_2_2_2"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E12:92_2_2_2"/>
<font color="Maroon" title="c10">C0</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:92_2_2_2"><span class="lab"><font color="Green" title="E61">A21</font></span></a>, <a class="txt" href="#E8:92_2_2_2"><span class="lab"><font color="Green" title="E63">A23</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:92"/>
<font color="Maroon" title="c5">C</font> is  <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:92"><span class="lab"><font color="Green" title="E52">A9</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>
