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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="chord.html#V6" title="CHORD:attr.6">chordal</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">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 <font color="Olive" title="b1">P</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  holds <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b4">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b4">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">P</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="Olive" title="b1">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Olive" title="b2">x</font>,<font color="Maroon" title="c1">G</font> ) holds <br/>( <font color="Olive" title="b1">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b3">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b3">e</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b3">e</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Maroon" title="c2">P</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  implies  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b3">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">G</font> ) holds <br/>( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:111"/><span class="lab"><font color="Green" title="E87">A1</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:111"/><span class="lab"><font color="Green" title="E88">A2</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</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">x</font>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b3">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">G</font> ) holds <br/>( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <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"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font> ) implies ( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:111"/><span class="lab"><font color="Green" title="E89">A3</font></span>: 
 not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">and </span><br/><a NAME="E4:111"/><span class="lab"><font color="Green" title="E90">A4</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="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E5:111"/><span class="lab"><font color="Green" title="E91">A5</font></span>: 
 for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</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">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">Q</font> = <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</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="#E1:111"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E89">A3</font></span></a>, <a class="txt" href="#E4:111"><span class="lab"><font color="Green" title="E90">A4</font></span></a>, <a class="ref" href="glib_001.html#T151" onmouseover="rs('glib_001/T151')" onmouseout="rh()">GLIB_001:151</a></span>;<br/>
<a NAME="E7:111"/><span class="lab"><font color="Green" title="E92">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</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="c2">P</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="#E4:111"><span class="lab"><font color="Green" title="E90">A4</font></span></a>, <a class="ref" href="glib_001.html#T64" onmouseover="rs('glib_001/T64')" onmouseout="rh()">GLIB_001:64</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1 &amp; 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  for <font color="Olive" title="b1">j</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 holds <br/> for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font> );<br/>
<div><span class="lab"><font color="Green" title="E93">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_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">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <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="c2">P</font> holds <br/><font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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="c2">P</font> implies <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:111_1"/><span class="lab"><font color="Green" title="E94">A8</font></span>: 
<font color="Maroon" title="c6">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="c2">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font></span><br/><a NAME="E2:111_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E3:111_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E94">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_1"/>
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111"><span class="lab"><font color="Green" title="E90">A4</font></span></a>, <a class="ref" href="glib_001.html#T65" onmouseover="rs('glib_001/T65')" onmouseout="rh()">GLIB_001:65</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="E9:111"/><span class="lab"><font color="Green" title="E94">A9</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111"><span class="lab"><font color="Green" title="E90">A4</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="E10:111"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">a</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:111_2"/><span class="lab"><font color="Green" title="E95">A10</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:111_2"/><span class="lab"><font color="Green" title="E96">A11</font></span>: 
4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E3:111_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font></span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 holds <br/> for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">j</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font></span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 implies  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:111_2"/><span class="lab"><font color="Green" title="E97">A12</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font></span>)</span> <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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font></span><br/>

<a NAME="E5:111_2"/>
<font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111_2"><span class="lab"><font color="Green" title="E96">A11</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E6:111_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_2"><span class="lab"><font color="Green" title="E97">A12</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E7:111_2"/><span class="lab"><font color="Green" title="E99">A14</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/>
<a NAME="E8:111_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A15</font></span>: 
<font color="Maroon" title="c7">j</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="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_2"><span class="lab"><font color="Green" title="E98">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:111_2"/><span class="lab"><font color="Green" title="E101">A16</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">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">  not <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">assume </span><a NAME="E10:111_2"/><span class="lab"><font color="Green" title="E102">A17</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_2_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="E1:111_2_1"/>
( <font color="Maroon" title="c7">j</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_2"><span class="lab"><font color="Green" title="E98">A13</font></span></a>, <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="111_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_2_1_1"/>
<font color="Maroon" title="c7">j</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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"> contradiction</span><br/><div class="add"><a NAME="E2:111_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E93">A7</font></span></a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><span class="kw">hence </span><a NAME="E3:111_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111"><span class="lab"><font color="Green" title="E91">A5</font></span></a>, <a class="txt" href="#E10:111_2"><span class="lab"><font color="Green" title="E102">A17</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="111_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_2_1_2"/><span class="lab"><font color="Green" title="E103">A18</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</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"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">lP2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 as   <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>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">jj</font> = <font color="Maroon" title="c7">j</font> as   <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">B</font> = <font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>);<br/><a NAME="E4:111_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E5:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A19</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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="#E8:111_2"><span class="lab"><font color="Green" title="E100">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E94">A9</font></span></a>, <a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><a NAME="E7:111_2_1_2"/><span class="lab"><font color="Green" title="E106">A21</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E5:111_2_1_2"><span class="lab"><font color="Green" title="E104">A19</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><div><span class="lab"><font color="Green" title="E107">A22</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_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">i</font> being   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">i</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">i</font> <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="c2">P</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <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="c2">P</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:111_2_1_2_1"/><span class="lab"><font color="Green" title="E108">A23</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <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="c2">P</font> )</span><br/><a NAME="E2:111_2_1_2_1"/>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_2_1_2_1"><span class="lab"><font color="Green" title="E108">A23</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E3:111_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A24</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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="#E7:111_2_1_2"><span class="lab"><font color="Green" title="E106">A21</font></span></a>, <a class="ref" href="chord.html#T3" target="_self" onmouseover="rs('chord/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E4:111_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a></span>;<br/><a NAME="E5:111_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A25</font></span>: 
<font color="Maroon" title="c12">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_2_1_2_1"><span class="lab"><font color="Green" title="E108">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:111_2_1_2_1"/><span class="lab"><font color="Green" title="E111">A26</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E7:111_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <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="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E5:111_2_1_2"><span class="lab"><font color="Green" title="E104">A19</font></span></a>, <a class="txt" href="#E5:111_2_1_2_1"><span class="lab"><font color="Green" title="E110">A25</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><a NAME="E8:111_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A27</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:111_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E5:111_2_1_2"><span class="lab"><font color="Green" title="E104">A19</font></span></a>, <a class="txt" href="#E6:111_2_1_2_1"><span class="lab"><font color="Green" title="E111">A26</font></span></a>, <a class="txt" href="#E5:111_2_1_2_1"><span class="lab"><font color="Green" title="E110">A25</font></span></a>, <a class="ref" href="glib_001.html#T36" onmouseover="rs('glib_001/T36')" onmouseout="rh()">GLIB_001:36</a></span>;<br/><span class="kw">hence </span><a NAME="E10:111_2_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> &amp; <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c12">i</font> <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="c2">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E93">A7</font></span></a>, <a class="txt" href="#E8:111_2_1_2_1"><span class="lab"><font color="Green" title="E112">A27</font></span></a>, <a class="txt" href="#E3:111_2_1_2_1"><span class="lab"><font color="Green" title="E109">A24</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</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">set </span><font color="Maroon" title="c12">C</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font>;<br/><a NAME="E9:111_2_1_2"/><span class="lab"><font color="Green" title="E108">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E5:111_2_1_2"><span class="lab"><font color="Green" title="E104">A19</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><a NAME="E10:111_2_1_2"/><span class="lab"><font color="Green" title="E109">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E5:111_2_1_2"><span class="lab"><font color="Green" title="E104">A19</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><a NAME="E11:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A30</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:111_2"><span class="lab"><font color="Green" title="E102">A17</font></span></a>, <a class="txt" href="#E6:111_2_1_2"><span class="lab"><font color="Green" title="E105">A20</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E111">A31</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_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">  for <font color="Olive" title="b1">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" 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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:111_2_1_2_2"/><span class="lab"><font color="Green" title="E112">A32</font></span>: 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font></span><br/><a NAME="E2:111_2_1_2_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E3:111_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_2_1_2_2"><span class="lab"><font color="Green" title="E112">A32</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_2_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:111_2_1_2"><span class="lab"><font color="Green" title="E110">A30</font></span></a>, <a class="ref" href="glib_001.html#T65" onmouseover="rs('glib_001/T65')" onmouseout="rh()">GLIB_001:65</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="E13:111_2_1_2"/><span class="lab"><font color="Green" title="E112">A33</font></span>: 
<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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_2_1_2"><span class="lab"><font color="Green" title="E103">A18</font></span></a>, <a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E14:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A34</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111_2_1_2"><span class="lab"><font color="Green" title="E106">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E15:111_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 5 <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="#E7:111_2_1_2"><span class="lab"><font color="Green" title="E106">A21</font></span></a>, <a class="txt" href="#E13:111_2_1_2"><span class="lab"><font color="Green" title="E112">A33</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E16:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A35</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:111_2"><span class="lab"><font color="Green" title="E102">A17</font></span></a>, <a class="txt" href="#E6:111_2_1_2"><span class="lab"><font color="Green" title="E105">A20</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a>, <a class="ref" href="glib_001.html#T64" onmouseover="rs('glib_001/T64')" onmouseout="rh()">GLIB_001:64</a></span>;<br/><div><span class="lab"><font color="Green" title="E115">A36</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_2_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"> contradiction</span><br/><a NAME="E2:111_2_1_2_3"/><span class="kw">then </span>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E3:111_2_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a></span>)</span></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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_2_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:111_2_1_2"><span class="lab"><font color="Green" title="E114">A35</font></span></a>, <a class="ref" href="glib_001.html#T112" onmouseover="rs('glib_001/T112')" onmouseout="rh()">GLIB_001:112</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="E18:111_2_1_2"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_2"><span class="lab"><font color="Green" title="E100">A15</font></span></a>, <a class="ref" href="glib_001.html#D8" onmouseover="rs('glib_001/D8')" onmouseout="rh()">GLIB_001:def 8</a></span>;<br/><a NAME="E19:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:111_2"><span class="lab"><font color="Green" title="E101">A16</font></span></a>, <a class="txt" href="#E6:111_2"><span class="lab"><font color="Green" title="E98">A13</font></span></a>, <a class="txt" href="#E7:111_2"><span class="lab"><font color="Green" title="E99">A14</font></span></a>, <a class="ref" href="glib_001.html#T89" onmouseover="rs('glib_001/T89')" onmouseout="rh()">GLIB_001:89</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E20:111_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E93">A7</font></span></a>, <a class="txt" href="#E6:111_2"><span class="lab"><font color="Green" title="E98">A13</font></span></a>, <a class="txt" href="#E7:111_2"><span class="lab"><font color="Green" title="E99">A14</font></span></a>, <a class="txt" href="#E9:111_2_1_2"><span class="lab"><font color="Green" title="E108">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E21:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A37</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</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="#E3:111"><span class="lab"><font color="Green" title="E89">A3</font></span></a>, <a class="txt" href="#E6:111_2_1_2"><span class="lab"><font color="Green" title="E105">A20</font></span></a></span>;<br/><a NAME="E22:111_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</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="#E10:111_2"><span class="lab"><font color="Green" title="E102">A17</font></span></a>, <a class="txt" href="#E14:111_2_1_2"><span class="lab"><font color="Green" title="E113">A34</font></span></a>, <a class="txt" href="#E10:111_2_1_2"><span class="lab"><font color="Green" title="E109">A29</font></span></a>, <a class="txt" href="#E6:111_2_1_2"><span class="lab"><font color="Green" title="E105">A20</font></span></a>, <a class="ref" href="chord.html#T33" target="_self" onmouseover="rs('chord/T33')" onmouseout="rh()">Th33</a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><a NAME="E23:111_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:111_2_1_2"><span class="lab"><font color="Green" title="E115">A36</font></span></a>, <a class="ref" href="chord.html#D11" target="_self" onmouseover="rs('chord/D11')" onmouseout="rh()">Def11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">m</font>, <font color="Maroon" title="c14">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E25:111_2_1_2"/><span class="lab"><font color="Green" title="E117">A38</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">and </span><br/><a NAME="E26:111_2_1_2"/><span class="lab"><font color="Green" title="E118">A39</font></span>: 
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:111_2_1_2"/><span class="lab"><font color="Green" title="E119">A40</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">and </span><br/><a NAME="E28:111_2_1_2"/><span class="lab"><font color="Green" title="E120">A41</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E29:111_2_1_2"/><span class="lab"><font color="Green" title="E121">A42</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  implies ( (  not <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> ) &amp; (  not <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c14">n</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T84" target="_self" onmouseover="rs('chord/T84')" onmouseout="rh()">Th83</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E31:111_2_1_2"/><span class="lab"><font color="Green" title="E122">A43</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:111_2_1_2"><span class="lab"><font color="Green" title="E120">A41</font></span></a></span>;<br/><a NAME="E32:111_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E33:111_2_1_2"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">m1</font> = <font color="Maroon" title="c16">m1</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E36:111_2_1_2"/><span class="lab"><font color="Green" title="E123">A44</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></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="#E10:111_2"><span class="lab"><font color="Green" title="E102">A17</font></span></a>, <a class="txt" href="#E6:111_2_1_2"><span class="lab"><font color="Green" title="E105">A20</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a>, <a class="ref" href="glib_001.html#T64" onmouseover="rs('glib_001/T64')" onmouseout="rh()">GLIB_001:64</a></span>;<br/><a NAME="E37:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></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="#E25:111_2_1_2"><span class="lab"><font color="Green" title="E117">A38</font></span></a>, <a class="txt" href="#E26:111_2_1_2"><span class="lab"><font color="Green" title="E118">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E38:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A45</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E39:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A46</font></span>: 
<font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E40:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A47</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_2_1_2"><span class="lab"><font color="Green" title="E107">A22</font></span></a></span>;<br/><a NAME="E41:111_2_1_2"/><span class="lab"><font color="Green" title="E127">A48</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font> <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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_2_1_2"><span class="lab"><font color="Green" title="E107">A22</font></span></a>, <a class="txt" href="#E39:111_2_1_2"><span class="lab"><font color="Green" title="E125">A46</font></span></a></span>;<br/><a NAME="E42:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A49</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E43:111_2_1_2"/><span class="lab"><font color="Green" title="E129">A50</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:111_2_1_2"><span class="lab"><font color="Green" title="E108">A28</font></span></a>, <a class="txt" href="#E11:111_2_1_2"><span class="lab"><font color="Green" title="E110">A30</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_2_1_2_4"/><span class="lab"><font color="Green" title="E130">A51</font></span>: 
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</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"> contradiction</span><br/><a NAME="E2:111_2_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>,<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:111_2_1_2"><span class="lab"><font color="Green" title="E111">A31</font></span></a>, <a class="txt" href="#E43:111_2_1_2"><span class="lab"><font color="Green" title="E129">A50</font></span></a>, <a class="txt" href="#E31:111_2_1_2"><span class="lab"><font color="Green" title="E122">A43</font></span></a>, <a class="txt" href="#E38:111_2_1_2"><span class="lab"><font color="Green" title="E124">A45</font></span></a>, <a class="txt" href="#E40:111_2_1_2"><span class="lab"><font color="Green" title="E126">A47</font></span></a></span>;<br/><a NAME="E3:111_2_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E93">A7</font></span></a>, <a class="txt" href="#E6:111_2"><span class="lab"><font color="Green" title="E98">A13</font></span></a>, <a class="txt" href="#E7:111_2"><span class="lab"><font color="Green" title="E99">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:111_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A52</font></span>: 
<font color="Maroon" title="c15">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>,<font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E5:111_2_1_2_4"/>
3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:111_2"><span class="lab"><font color="Green" title="E102">A17</font></span></a>, <a class="txt" href="#E14:111_2_1_2"><span class="lab"><font color="Green" title="E113">A34</font></span></a>, <a class="txt" href="#E10:111_2_1_2"><span class="lab"><font color="Green" title="E109">A29</font></span></a>, <a class="txt" href="#E6:111_2_1_2"><span class="lab"><font color="Green" title="E105">A20</font></span></a>, <a class="txt" href="#E21:111_2_1_2"><span class="lab"><font color="Green" title="E116">A37</font></span></a>, <a class="txt" href="#E29:111_2_1_2"><span class="lab"><font color="Green" title="E121">A42</font></span></a>, <a class="txt" href="#E1:111_2_1_2_4"><span class="lab"><font color="Green" title="E130">A51</font></span></a>, <a class="ref" href="chord.html#T7" target="_self" onmouseover="rs('chord/T7')" onmouseout="rh()">Th7</a>, <a class="ref" href="chord.html#T33" target="_self" onmouseover="rs('chord/T33')" onmouseout="rh()">Th33</a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:111_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A53</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E7:111_2_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A54</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E8:111_2_1_2_4"/>
<font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_2_1_2_4"><span class="lab"><font color="Green" title="E132">A53</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E9:111_2_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="txt" href="#E42:111_2_1_2"><span class="lab"><font color="Green" title="E128">A49</font></span></a>, <a class="txt" href="#E4:111_2_1_2_4"><span class="lab"><font color="Green" title="E131">A52</font></span></a>, <a class="txt" href="#E7:111_2_1_2_4"><span class="lab"><font color="Green" title="E133">A54</font></span></a>, <a class="ref" href="chord.html#T92" target="_self" onmouseover="rs('chord/T92')" onmouseout="rh()">Th91</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="E45:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:111_2_1_2"><span class="lab"><font color="Green" title="E123">A44</font></span></a>, <a class="txt" href="#E26:111_2_1_2"><span class="lab"><font color="Green" title="E118">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E46:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A55</font></span>: 
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T3" target="_self" onmouseover="rs('chord/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E47:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:111_2_1_2"><span class="lab"><font color="Green" title="E111">A31</font></span></a></span>;<br/><a NAME="E48:111_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E49:111_2_1_2"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">n1</font> = <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">n1</font> = <font color="Maroon" title="c18">n1</font> as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E52:111_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:111_2_1_2"><span class="lab"><font color="Green" title="E117">A38</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E53:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E54:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font>
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_5"><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">  not <font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span></a><div class="add"><a NAME="E1:111_2_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:111_2_1_2"><span class="lab"><font color="Green" title="E111">A31</font></span></a>, <a class="txt" href="#E38:111_2_1_2"><span class="lab"><font color="Green" title="E124">A45</font></span></a></span>;<br/><a NAME="E2:111_2_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A58</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E93">A7</font></span></a>, <a class="txt" href="#E40:111_2_1_2"><span class="lab"><font color="Green" title="E126">A47</font></span></a>, <a class="txt" href="#E42:111_2_1_2"><span class="lab"><font color="Green" title="E128">A49</font></span></a></span>;<br/><a NAME="E3:111_2_1_2_5"/><span class="lab"><font color="Green" title="E134">A59</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E4:111_2_1_2_5"/>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:111_2_1_2"><span class="lab"><font color="Green" title="E127">A48</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">kk</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:111_2_1_2_5"/><span class="lab"><font color="Green" title="E135">A60</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span> <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="c2">P</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="#E3:111_2_1_2_5"><span class="lab"><font color="Green" title="E134">A59</font></span></a>, <a class="txt" href="chord.html#E15"><span class="lab"><font color="Green" title="E14">Lm2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">assume </span><a NAME="E7:111_2_1_2_5"/><span class="lab"><font color="Green" title="E136">A61</font></span>: 
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</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"> contradiction</span><br/><div><span class="lab"><font color="Green" title="E137">A62</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_5_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">  not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_2_1_2_5_1"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:111_2_1_2_5_1"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:111_2_1_2_5_1"/><span class="kw">then </span>
( 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T11" target="_self" onmouseover="rs('chord/T11')" onmouseout="rh()">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_2_1_2_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111_2_1_2"><span class="lab"><font color="Green" title="E106">A21</font></span></a>, <a class="txt" href="#E25:111_2_1_2"><span class="lab"><font color="Green" title="E117">A38</font></span></a>, <a class="txt" href="#E27:111_2_1_2"><span class="lab"><font color="Green" title="E119">A40</font></span></a>, <a class="txt" href="#E7:111_2_1_2_5"><span class="lab"><font color="Green" title="E136">A61</font></span></a>, <a class="txt" href="#E6:111_2_1_2_5"><span class="lab"><font color="Green" title="E135">A60</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:111_2_1_2_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E10:111_2_1_2_5"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:111_2_1_2"><span class="lab"><font color="Green" title="E113">A34</font></span></a>, <a class="txt" href="#E21:111_2_1_2"><span class="lab"><font color="Green" title="E116">A37</font></span></a>, <a class="txt" href="#E11:111_2_1_2"><span class="lab"><font color="Green" title="E110">A30</font></span></a>, <a class="txt" href="#E36:111_2_1_2"><span class="lab"><font color="Green" title="E123">A44</font></span></a>, <a class="txt" href="#E29:111_2_1_2"><span class="lab"><font color="Green" title="E121">A42</font></span></a>, <a class="txt" href="#E7:111_2_1_2_5"><span class="lab"><font color="Green" title="E136">A61</font></span></a>, <a class="ref" href="chord.html#T33" target="_self" onmouseover="rs('chord/T33')" onmouseout="rh()">Th33</a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E11:111_2_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A63</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><div><span class="lab"><font color="Green" title="E139">A64</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_5_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">  not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_2_1_2_5_2"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:111_2_1_2_5_2"/><span class="kw">then </span>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:111_2_1_2_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_2_1_2_5"><span class="lab"><font color="Green" title="E135">A60</font></span></a>, <a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</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="lab"><font color="Green" title="E140">A65</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_2_1_2_5_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_2_1_2_5_3"/>
<font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:111_2_1_2_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A66</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E3:111_2_1_2_5_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">kk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:111_2_1_2_5"><span class="lab"><font color="Green" title="E138">A63</font></span></a>, <a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_2_1_2_5_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_2"><span class="lab"><font color="Green" title="E97">A12</font></span></a>, <a class="txt" href="#E6:111_2_1_2_5"><span class="lab"><font color="Green" title="E135">A60</font></span></a>, <a class="txt" href="#E2:111_2_1_2_5_3"><span class="lab"><font color="Green" title="E141">A66</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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><a NAME="E14:111_2_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_2_1_2"><span class="lab"><font color="Green" title="E105">A20</font></span></a>, <a class="txt" href="#E12:111_2_1_2"><span class="lab"><font color="Green" title="E111">A31</font></span></a>, <a class="txt" href="#E7:111_2_1_2_5"><span class="lab"><font color="Green" title="E136">A61</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:111_2_1_2_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_2"><span class="lab"><font color="Green" title="E95">A10</font></span></a>, <a class="txt" href="#E31:111_2_1_2"><span class="lab"><font color="Green" title="E122">A43</font></span></a>, <a class="txt" href="#E6:111_2_1_2_5"><span class="lab"><font color="Green" title="E135">A60</font></span></a>, <a class="txt" href="#E8:111_2_1_2_5"><span class="lab"><font color="Green" title="E137">A62</font></span></a>, <a class="txt" href="#E12:111_2_1_2_5"><span class="lab"><font color="Green" title="E139">A64</font></span></a>, <a class="txt" href="#E13:111_2_1_2_5"><span class="lab"><font color="Green" title="E140">A65</font></span></a>, <a class="txt" href="#E2:111_2_1_2_5"><span class="lab"><font color="Green" title="E133">A58</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E56:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:111_2_1_2"><span class="lab"><font color="Green" title="E130">A55</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E57:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A67</font></span>: 
<font color="Maroon" title="c19">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E58:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font> <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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_2_1_2"><span class="lab"><font color="Green" title="E107">A22</font></span></a></span>;<br/><a NAME="E59:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A68</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</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="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E60:111_2_1_2"/>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E61:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:111_2_1_2"><span class="lab"><font color="Green" title="E117">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E62:111_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E63:111_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A69</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E64:111_2_1_2"/><span class="lab"><font color="Green" title="E136">A70</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c8">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:111_2_1_2"><span class="lab"><font color="Green" title="E111">A31</font></span></a>, <a class="txt" href="#E38:111_2_1_2"><span class="lab"><font color="Green" title="E124">A45</font></span></a></span>;<br/><a NAME="E65:111_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c10">jj</font>,<font color="Maroon" title="c9">lP2</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_2_1_2"><span class="lab"><font color="Green" title="E107">A22</font></span></a>, <a class="txt" href="#E57:111_2_1_2"><span class="lab"><font color="Green" title="E133">A67</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E66:111_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="txt" href="#E31:111_2_1_2"><span class="lab"><font color="Green" title="E122">A43</font></span></a>, <a class="txt" href="#E40:111_2_1_2"><span class="lab"><font color="Green" title="E126">A47</font></span></a>, <a class="txt" href="#E59:111_2_1_2"><span class="lab"><font color="Green" title="E134">A68</font></span></a>, <a class="txt" href="#E64:111_2_1_2"><span class="lab"><font color="Green" title="E136">A70</font></span></a>, <a class="txt" href="#E47:111_2_1_2"><span class="lab"><font color="Green" title="E131">A56</font></span></a>, <a class="txt" href="#E63:111_2_1_2"><span class="lab"><font color="Green" title="E135">A69</font></span></a>, <a class="txt" href="#E54:111_2_1_2"><span class="lab"><font color="Green" title="E132">A57</font></span></a>, <a class="ref" href="chord.html#T92" target="_self" onmouseover="rs('chord/T92')" onmouseout="rh()">Th91</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="E11:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A71</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">a</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<a NAME="E12:111"/><span class="lab"><font color="Green" title="E96">A72</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" onmouseover="rs('nat_1/S4')" onmouseout="rh()">NAT_1:sch 4</a>(<a class="txt" href="#E11:111"><span class="lab"><font color="Green" title="E95">A71</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E97">A73</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 holds <br/> for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 implies  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:111_3"/><span class="lab"><font color="Green" title="E98">A74</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font></span><br/><a NAME="E2:111_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:111_3"/><span class="lab"><font color="Green" title="E99">A75</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <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="c2">P</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="#E1:111_3"><span class="lab"><font color="Green" title="E98">A74</font></span></a>, <a class="txt" href="chord.html#E15"><span class="lab"><font color="Green" title="E14">Lm2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E100">A76</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_3_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">  not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_3_1"/>
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:111_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K1" title="NAT_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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:111_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_3"><span class="lab"><font color="Green" title="E99">A75</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_3_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">  not 2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_3_2"/><span class="lab"><font color="Green" title="E101">A77</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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"> contradiction</span><br/><a NAME="E2:111_3_2"/>
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_3"><span class="lab"><font color="Green" title="E98">A74</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:111_3_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_3"><span class="lab"><font color="Green" title="E99">A75</font></span></a>, <a class="txt" href="#E1:111_3_2"><span class="lab"><font color="Green" title="E101">A77</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</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">hence </span><a NAME="E7:111_3"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:111"><span class="lab"><font color="Green" title="E96">A72</font></span></a>, <a class="txt" href="#E4:111_3"><span class="lab"><font color="Green" title="E99">A75</font></span></a>, <a class="txt" href="#E5:111_3"><span class="lab"><font color="Green" title="E100">A76</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="lab"><font color="Green" title="E98">A78</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">Q</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:111_4"/>
<font color="Maroon" title="c5">Q</font> is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:111_4"/><span class="lab"><font color="Green" title="E99">A79</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E4:111_4"/><span class="lab"><font color="Green" title="E100">A80</font></span>: 
<font color="Maroon" title="c7">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="c5">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:111_4"/>
<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">and </span><br/><a NAME="E6:111_4"/><span class="lab"><font color="Green" title="E101">A81</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E7:111_4"/>
( <font color="Maroon" title="c5">Q</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  implies ( (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c7">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="c5">Q</font> ) &amp; (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c7">n</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="c5">Q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c7">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="c5">Q</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T84" target="_self" onmouseover="rs('chord/T84')" onmouseout="rh()">Th83</a></span>;<br/><a NAME="E8:111_4"/>
<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</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="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E3:111_4"><span class="lab"><font color="Green" title="E99">A79</font></span></a>, <a class="txt" href="#E4:111_4"><span class="lab"><font color="Green" title="E100">A80</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:111_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A82</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E10:111_4"/>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><a NAME="E11:111_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A83</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_4"><span class="lab"><font color="Green" title="E99">A79</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_4_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="E1:111_4_1"/>
( <font color="Maroon" title="c7">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="c5">Q</font> or <font color="Maroon" title="c7">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="c5">Q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_4"><span class="lab"><font color="Green" title="E100">A80</font></span></a>, <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="111_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_4_1_1"/><span class="lab"><font color="Green" title="E104">A84</font></span>: 
<font color="Maroon" title="c7">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="c5">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:111_4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E3:111_4"><span class="lab"><font color="Green" title="E99">A79</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:111_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E94">A9</font></span></a>, <a class="txt" href="#E13:111"><span class="lab"><font color="Green" title="E97">A73</font></span></a>, <a class="txt" href="#E6:111_4"><span class="lab"><font color="Green" title="E101">A81</font></span></a>, <a class="txt" href="#E1:111_4_1_1"><span class="lab"><font color="Green" title="E104">A84</font></span></a>, <a class="ref" href="chord.html#T3" target="_self" onmouseover="rs('chord/T3')" onmouseout="rh()">Th3</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="111_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_4_1_2"/><span class="lab"><font color="Green" title="E104">A85</font></span>: 
<font color="Maroon" title="c7">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="c5">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:111_4_1_2"/><span class="lab"><font color="Green" title="E105">A86</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E93">A7</font></span></a>, <a class="txt" href="#E9:111_4"><span class="lab"><font color="Green" title="E102">A82</font></span></a></span>;<br/><a NAME="E3:111_4_1_2"/><span class="lab"><font color="Green" title="E106">A87</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c2">P</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <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="#E7:111"><span class="lab"><font color="Green" title="E92">A6</font></span></a>, <a class="txt" href="#E1:111_4_1_2"><span class="lab"><font color="Green" title="E104">A85</font></span></a>, <a class="ref" href="chord.html#T3" target="_self" onmouseover="rs('chord/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E4:111_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E93">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:111_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E87">A1</font></span></a>, <a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E88">A2</font></span></a>, <a class="txt" href="#E3:111_4"><span class="lab"><font color="Green" title="E99">A79</font></span></a>, <a class="txt" href="#E6:111_4"><span class="lab"><font color="Green" title="E101">A81</font></span></a>, <a class="txt" href="#E11:111_4"><span class="lab"><font color="Green" title="E103">A83</font></span></a>, <a class="txt" href="#E3:111_4_1_2"><span class="lab"><font color="Green" title="E106">A87</font></span></a>, <a class="txt" href="#E2:111_4_1_2"><span class="lab"><font color="Green" title="E105">A86</font></span></a>, <a class="ref" href="chord.html#T92" target="_self" onmouseover="rs('chord/T92')" onmouseout="rh()">Th91</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="E15:111"/>
<font color="Maroon" title="c5">Q</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="c2">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111"><span class="lab"><font color="Green" title="E90">A4</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/>
<a NAME="E16:111"/><span class="kw">then </span>
<font color="Maroon" title="c5">Q</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T88" onmouseover="rs('glib_001/T88')" onmouseout="rh()">GLIB_001:88</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:111"/>
( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c4">e</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E89">A3</font></span></a>, <a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E94">A9</font></span></a>, <a class="txt" href="#E14:111"><span class="lab"><font color="Green" title="E98">A78</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>
