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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font>, <font color="Olive" title="b3">v3</font>, <font color="Olive" title="b4">v4</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</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; <font color="Olive" title="b1">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v3</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">v3</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">v4</font> &amp; <font color="Olive" title="b3">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">v4</font> &amp; <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b2">v2</font>,<font color="Olive" title="b3">v3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Olive" title="b3">v3</font>,<font color="Olive" title="b4">v4</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  holds <br/> ex <font color="Olive" title="b5">P</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 &amp; <font color="Olive" title="b5">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b5">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Olive" title="b3">v3</font>,<font color="Olive" title="b4">v4</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Olive" title="b5">P</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="Olive" title="b1">v1</font> &amp; <font color="Olive" title="b5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v3</font> &amp; <font color="Olive" title="b5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v4</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c4">v3</font>, <font color="Maroon" title="c5">v4</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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; <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font> &amp; <font color="Maroon" title="c4">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font> &amp; <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  &amp; <font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a>  implies  ex <font color="Olive" title="b1">P</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Olive" title="b1">P</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="c2">v1</font> &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53"/><span class="lab"><font color="Green" title="E46">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>
 <span class="kw">and </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E47">A2</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="c4">v3</font>
 <span class="kw">and </span><br/><a NAME="E3:53"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v3</font>
 <span class="kw">and </span><br/><a NAME="E4:53"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font>
 <span class="kw">and </span><br/><a NAME="E5:53"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c4">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font>
 <span class="kw">and </span><br/><a NAME="E6:53"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">and </span><br/><a NAME="E7:53"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 <span class="kw">and </span><br/><a NAME="E8:53"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font> <a href="chord.html#R1" title="CHORD:pred.1">are_adjacent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Olive" title="b1">P</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="c2">v1</font> &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">R</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c7">e1</font>, <font color="Maroon" title="c8">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:53"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
not <font color="Maroon" title="c6">R</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">and </span><br/><a NAME="E11:53"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 <span class="kw">and </span><br/><a NAME="E12:53"/>
<font color="Maroon" title="c6">R</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">and </span><br/><a NAME="E13:53"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c7">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E14:53"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c8">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E15:53"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
<font color="Maroon" title="c6">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c8">e2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">and </span><br/><a NAME="E16:53"/><span class="lab"><font color="Green" title="E59">A14</font></span>: 
<font color="Maroon" title="c6">R</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E17:53"/><span class="lab"><font color="Green" title="E60">A15</font></span>: 
<font color="Maroon" title="c6">R</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="c2">v1</font>
 <span class="kw">and </span><br/><a NAME="E18:53"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
<font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font>
 <span class="kw">and </span><br/><a NAME="E19:53"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
<font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E7:53"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="chord.html#T47" target="_self" title="CHORD:th.47">Th47</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E21:53"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<font color="Maroon" title="c9">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="chord.html#D3" target="_self" title="CHORD:def.3">Def3</a></span>;<br/>
<a NAME="E22:53"/><span class="lab"><font color="Green" title="E64">A19</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">R</font> holds <br/><font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c10">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="c6">R</font> implies <font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E65">A20</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">and </span><br/><a NAME="E2:53_1"/><span class="lab"><font color="Green" title="E66">A21</font></span>: 
<font color="Maroon" title="c10">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="c6">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:53_1_1"/>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="chord.html#T8" target="_self" title="CHORD:th.8">Th8</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:53_1_1_1"/>
<font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E18:53"><span class="lab"><font color="Green" title="E61">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:53_1_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:53_1_1_2"/>
<font color="Maroon" title="c6">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E23:53"/><span class="lab"><font color="Green" title="E65">A22</font></span>: 
<font color="Maroon" title="c8">e2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E14:53"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">P</font> = <font color="Maroon" title="c6">R</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e3</font>;<br/>
<a NAME="E24:53"/>
<font color="Maroon" title="c7">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E13:53"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></span>;<br/>
<a NAME="E25:53"/><span class="kw">then </span>
not <font color="Maroon" title="c9">e3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">R</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:53"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E23:53"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">P</font> = <font color="Maroon" title="c6">R</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c9">e3</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E10:53"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E22:53"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="glib_001.html#T151" title="GLIB_001:th.151">GLIB_001:151</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 &amp; <font color="Maroon" title="c11">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Maroon" title="c11">P</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="c2">v1</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<a NAME="E27:53"/><span class="lab"><font color="Green" title="E66">A23</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="glib_001.html#T65" title="GLIB_001:th.65">GLIB_001:65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:53"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Maroon" title="c11">P</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="c2">v1</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<a NAME="E29:53"/>
7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:53"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="ref" href="glib_001.html#T113" title="GLIB_001:th.113">GLIB_001:113</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:53"/>
<font color="Maroon" title="c11">P</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Maroon" title="c11">P</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="c2">v1</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<a NAME="E31:53"/>
<font color="Maroon" title="c11">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</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"><font color="Maroon" title="c5">v4</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="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E16:53"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="glib_001.html#T96" title="GLIB_001:th.96">GLIB_001:96</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:53"/>
<font color="Maroon" title="c11">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T46" title="ENUMSET1:th.46">ENUMSET1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">P</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="c2">v1</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<a NAME="E33:53"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E34:53"/>
<font color="Maroon" title="c11">P</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="c2">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E17:53"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<a NAME="E35:53"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:53"/>
<font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E18:53"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> )</span><br/>

<a NAME="E37:53"/>
5 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:53"/>
<font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="glib_001.html#T66" title="GLIB_001:th.66">GLIB_001:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font></span><br/>

<a NAME="E39:53"/>
<font color="Maroon" title="c11">P</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="c5">v4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E19:53"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="ref" href="glib_001.html#T64" title="GLIB_001:th.64">GLIB_001:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:53"/>
<font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 7 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:53"><span class="lab"><font color="Green" title="E66">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
