<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:48"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">X</font> = <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="Olive" title="b1">P</font></span>)</span> where <font color="Olive" title="b1">P</font> is   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:48"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">a</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E4:48"><i><font color="Green" title="E42">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:48"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">b</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="chord.html#E7:48"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E10:48"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c9">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="glib_002.html#D1" title="GLIB_002:def.1">GLIB_002:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">W2</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E12:48"/><i><font color="Green" title="E45">A6</font></i>: 
( <font color="Maroon" title="c10">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">W</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c9">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  &amp; <font color="Maroon" title="c10">W2</font> is <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> )
 <b>by </b><i><a class="ref" href="chord.html#T39" target="_self" title="CHORD:th.39">Th39</a></i>;<br/>
<a NAME="E13:48"/>
( <font color="Maroon" title="c9">W</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="c6">a</font> &amp; <font color="Maroon" title="c9">W</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="c8">b</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E10:48"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E14:48"/><b>then </b>
( <font color="Maroon" title="c10">W2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">W2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="chord.html#E4:48"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="chord.html#E7:48"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="chord.html#E12:48"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E15:48"/><b>then </b><i><font color="Green" title="E46">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">W2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b1">P</font></span>)</span> where <font color="Olive" title="b1">P</font> is   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span>}</span> 
 
;<br/>
<a NAME="E16:48"/>
<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="Olive" title="b1">P</font></span>)</span> where <font color="Olive" title="b1">P</font> is   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="Olive" title="b1">P</font></span>)</span> where <font color="Olive" title="b1">P</font> is   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
 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/>( <font color="Maroon" title="c11">x</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="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 
;<br/>
<b>hence </b><a NAME="E3:48_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">X</font> = <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="Olive" title="b1">P</font></span>)</span> where <font color="Olive" title="b1">P</font> is   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="chord.html#E15:48"><i><font color="Green" title="E46">A7</font></i></a></i>;<br/>
<a NAME="E18:48"/>
 <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">M</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E20:48"/><i><font color="Green" title="E47">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c11">X</font> &amp; <font color="Maroon" title="c12">M</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c12">M</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2"/>
<font color="Maroon" title="c12">M</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="c12">M</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><a NAME="E2:48_2"/><b>then </b>
<font color="Maroon" title="c12">M</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:48_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:48"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:48"/><b>then </b><i><font color="Green" title="E48">A9</font></i>: 
not <font color="Maroon" title="c12">M</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">P2</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:48_3"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon" title="c13">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c12">M</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c12">M</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><a NAME="E2:48_3"/>
( <font color="Maroon" title="c13">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c13">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="chord.html#E1:48_3"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E3:48_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">P2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><b>hence </b><a NAME="E4:48_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">P2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M</font>
 <b>by </b><i><a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E24:48"/><b>then </b><i><font color="Green" title="E49">A11</font></i>: 
<font color="Maroon" title="c12">M</font> is <a href="chord.html#V1" title="CHORD:attr.1">minlength</a>
 
<b>by </b><i><a class="ref" href="chord.html#T38" target="_self" title="CHORD:th.38">Th38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:48_4"/><i><font color="Green" title="E50">A12</font></i>: 
( 1 <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#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="c12">M</font> )
 ;<br/><b>assume </b><a NAME="E2:48_4"/><i><font color="Green" title="E51">A13</font></i>: 
( <font color="Maroon" title="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c14">nn</font> = <font color="Maroon" title="c13">n</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_4_1"/>
( <font color="Maroon" title="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 </span><b>by </b><i><a class="txt" href="chord.html#E2:48_4"><i><font color="Green" title="E51">A13</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1"><b>suppose </b></a><a NAME="E1:48_4_1_1"/><i><font color="Green" title="E52">A14</font></i>: 
<font color="Maroon" title="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c15">T</font> = <font color="Maroon" title="c12">M</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c14">nn</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M</font></span>)</span> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:48_4_1_1"/>
( <font color="Maroon" title="c15">T</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c15">T</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="chord.html#E1:48_4_1_1"><i><font color="Green" title="E52">A14</font></i></a>, <a class="txt" href="chord.html#E1:48_4"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E4:48_4_1_1"/><b>then </b><i><font color="Green" title="E53">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 ;<br/><a NAME="E5:48_4_1_1"/><i><font color="Green" title="E54">A16</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">T</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">nn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E1:48_4"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><a NAME="E6:48_4_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">T</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="chord.html#E1:48_4"><i><font color="Green" title="E50">A12</font></i></a>, <a class="txt" href="chord.html#E5:48_4_1_1"><i><font color="Green" title="E54">A16</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:48_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E4:48_4_1_1"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_2"><b>suppose </b></a><a NAME="E1:48_4_1_2"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c15">T</font> = <font color="Maroon" title="c12">M</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c14">nn</font> as   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E3:48_4_1_2"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c12">M</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:48_4"><i><font color="Green" title="E50">A12</font></i></a></i>;<br/><a NAME="E4:48_4_1_2"/><b>then </b>
( <font color="Maroon" title="c15">T</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="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c15">T</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="c12">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">nn</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E5:48_4_1_2"/><b>then </b><i><font color="Green" title="E53">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X</font>
 <b>by </b><i><a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="chord.html#E1:48_4_1_2"><i><font color="Green" title="E52">A17</font></i></a></i>;<br/><a NAME="E6:48_4_1_2"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="c12">M</font> )
 <b>by </b><i><a class="txt" href="chord.html#E1:48_4"><i><font color="Green" title="E50">A12</font></i></a></i>;<br/><a NAME="E7:48_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">T</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">nn</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></i>;<br/><b>hence </b><a NAME="E8:48_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:48_4_1_2"><i><font color="Green" title="E53">A18</font></i></a>, <a class="txt" href="chord.html#E1:48_4"><i><font color="Green" title="E50">A12</font></i></a>, <a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E26:48"/>
 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/>( <font color="Olive" title="b1">P</font> is <a href="chord.html#V1" title="CHORD:attr.1">minlength</a> &amp; not <font color="Olive" title="b1">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; ( for <font color="Olive" title="b2">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 1 <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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font> holds <br/>( not <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) ) )
 <b>by </b><i><a class="txt" href="chord.html#E22:48"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="chord.html#E20:48"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="chord.html#E24:48"><i><font color="Green" title="E49">A11</font></i></a></i>;<br/>


</div>
