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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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  ex <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">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="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">P</font> &amp;  ex <font color="Olive" title="b4">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b4">e</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> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font>,<font color="Maroon" title="c1">G</font> &amp; ( <font color="Olive" title="b1">P</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b3">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="Olive" title="b1">P</font> ) &amp; (  not <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b3">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="Olive" title="b1">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Olive" title="b3">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="Olive" title="b1">P</font> ) ) ) ) holds <br/><font color="Olive" title="b1">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a></span><br/><b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">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="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp;  ex <font color="Olive" title="b3">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b3">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="Olive" title="b1">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Maroon" title="c1">G</font> &amp; ( <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> ) &amp; (  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Olive" title="b2">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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> ) ) ) ) implies <font color="Maroon" title="c2">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> )</span><br/>



<b>given </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:101"/><i><font color="Green" title="E82">A1</font></i>: 
<font color="Maroon" title="c3">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="c4">n</font>
 <b>and </b><br/><a NAME="E3:101"/><i><font color="Green" title="E83">A2</font></i>: 
<font color="Maroon" title="c4">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>
 <b>and </b><br/><a NAME="E4:101"/><i><font color="Green" title="E84">A3</font></i>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E5:101"/><i><font color="Green" title="E85">A4</font></i>: 
( <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> implies ( (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c4">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="c2">P</font> ) &amp; (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c4">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="c2">P</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 ) &amp; (  not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or  not <font color="Maroon" title="c4">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="c2">P</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a></span><br/>

<b>take </b>
<font color="Maroon" title="c3">m</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="chord.html#D10" target="_self" title="CHORD:def.10">CHORD:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c3">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="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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> &amp;  ex <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<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>,<font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b2">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> <font color="Maroon" title="c3">m</font>,<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>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c4">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">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="c4">n</font> &amp; <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp;  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> 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> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<a NAME="E6:101"/><i><font color="Green" title="E86">A5</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:101"/><i><font color="Green" title="E87">A6</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="chord.html#E2:101"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<b>thus </b><a NAME="E8:101"/>
<font color="Maroon" title="c3">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="c4">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E2:101"><i><font color="Green" title="E82">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp;  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> 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> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<b>thus </b><a NAME="E9:101"/>
<font color="Maroon" title="c4">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>
 <b>by </b><i><a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> &amp;  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> 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> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span><br/>

<a NAME="E10:101"/><i><font color="Green" title="E88">A7</font></i>: 
<font color="Maroon" title="c3">m</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="c2">P</font>
 
<b>by </b><i><a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="chord.html#E7:101"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:101_1"/><b>then </b>
<font color="Maroon" title="c3">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> 1
 <b>by </b><i><a class="txt" href="chord.html#E2:101"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:101_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E12:101"/><b>then </b><i><font color="Green" title="E89">A8</font></i>: 
not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</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">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> 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> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> ) )</span>
<div class="add"><b>assume </b><a NAME="E1:101_2"/><i><font color="Green" title="E90">A9</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:101_2"/><b>then </b><i><font color="Green" title="E91">A10</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">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="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E7:101"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_2"/>
( <font color="Maroon" title="c2">P</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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> )
 ;<br/><a NAME="E4:101_2"/><b>then </b>
<font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E1:101_2"><i><font color="Green" title="E90">A9</font></i></a>, <a class="txt" href="chord.html#E2:101_2"><i><font color="Green" title="E91">A10</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E5:101_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="chord.html#E5:101"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E7:101"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="chord.html#E12:101"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="chord.html#E1:101_2"><i><font color="Green" title="E90">A9</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E14:101"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="chord.html#E4:101"><i><font color="Green" title="E84">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> 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> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font></span><br/>

<b>let </b><font color="Maroon" title="c5">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  implies  not <font color="Maroon" title="c5">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> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E15:101"/><i><font color="Green" title="E90">A11</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>and </b><br/><a NAME="E16:101"/><i><font color="Green" title="E91">A12</font></i>: 
<font color="Maroon" title="c5">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> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c6">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E18:101"/><i><font color="Green" title="E92">A13</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN: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="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <b>and </b><br/><a NAME="E19:101"/><i><font color="Green" title="E93">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="chord.html#E15:101"><i><font color="Green" title="E90">A11</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:101"/><i><font color="Green" title="E94">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</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="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="chord.html#E18:101"><i><font color="Green" title="E92">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E22:101"/><b>then </b><i><font color="Green" title="E95">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</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">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D15" title="GLIB_001:def.15">GLIB_001:def 15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">k</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<a NAME="E23:101"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font>
 
<b>by </b><i><a class="txt" href="chord.html#E21:101"><i><font color="Green" title="E94">A15</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E24:101"/><b>then </b>
2 <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> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E25:101"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">k</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E27:101"/><i><font color="Green" title="E96">A17</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font>
 
;<br/>
<a NAME="E28:101"/><i><font color="Green" title="E97">A18</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E29:101"/>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="chord.html#E18:101"><i><font color="Green" title="E92">A13</font></i></a>, <a class="ref" href="glib_001.html#T79" title="GLIB_001:th.79">GLIB_001:79</a></i>;<br/>
<a NAME="E30:101"/><b>then </b>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">i</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>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E31:101"/><b>then </b><i><font color="Green" title="E98">A19</font></i>: 
<font color="Maroon" title="c9">k</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="c2">P</font>
 
<b>by </b><i><a class="txt" href="chord.html#E27:101"><i><font color="Green" title="E96">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E32:101"/><b>then </b><i><font color="Green" title="E99">A20</font></i>: 
<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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c9">k</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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="chord.html#E28:101"><i><font color="Green" title="E97">A18</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/>
<a NAME="E33:101"/><i><font color="Green" title="E100">A21</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 
<b>by </b><i><a class="txt" href="chord.html#E31:101"><i><font color="Green" title="E98">A19</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3"/>
( ( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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="c3">m</font> &amp; <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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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="c4">n</font> &amp; <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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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> <font color="Maroon" title="c3">m</font> ) )
 </span><b>by </b><i><a class="txt" href="chord.html#E16:101"><i><font color="Green" title="E91">A12</font></i></a>, <a class="txt" href="chord.html#E19:101"><i><font color="Green" title="E93">A14</font></i></a>, <a class="txt" href="chord.html#E22:101"><i><font color="Green" title="E95">A16</font></i></a>, <a class="txt" href="chord.html#E32:101"><i><font color="Green" title="E99">A20</font></i></a>, <a class="ref" href="glib_000.html#T18" title="GLIB_000:th.18">GLIB_000:18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1"><b>suppose </b></a><a NAME="E1:101_3_1"/><i><font color="Green" title="E101">A22</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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="c3">m</font> &amp; <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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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> <font color="Maroon" title="c4">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_1"><b>suppose </b></a><a NAME="E1:101_3_1_1_1"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:101_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E10:101"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="chord.html#E28:101"><i><font color="Green" title="E97">A18</font></i></a>, <a class="txt" href="chord.html#E1:101_3_1"><i><font color="Green" title="E101">A22</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2"><b>suppose </b></a><a NAME="E1:101_3_1_1_2"/><i><font color="Green" title="E102">A23</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_1_1_2_1"/>
( <font color="Maroon" title="c9">k</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="c4">n</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c9">k</font> <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> <font color="Maroon" title="c4">n</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:101_3_1_1_2_1_1"/>
<font color="Maroon" title="c9">k</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="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E103">A24</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E1:101_3_1"><i><font color="Green" title="E101">A22</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_3_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E4:101_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">k</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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E2:101_3_1_1_2_1_1"><i><font color="Green" title="E103">A24</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E5:101_3_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:101_3_1_1_2_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:101_3_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:101"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="chord.html#E1:101_3_1_1_2"><i><font color="Green" title="E102">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:101_3_1_1_2_1_3"/><i><font color="Green" title="E103">A25</font></i>: 
<font color="Maroon" title="c9">k</font> <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> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_3_1_1_2_1_3"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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>
 <b>by </b><i><a class="txt" href="chord.html#E31:101"><i><font color="Green" title="E98">A19</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E3:101_3_1_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E1:101_3_1"><i><font color="Green" title="E101">A22</font></i></a>, <a class="txt" href="chord.html#E1:101_3_1_1_2_1_3"><i><font color="Green" title="E103">A25</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E4:101_3_1_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E2:101"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1_1_3"><b>suppose </b></a><a NAME="E1:101_3_1_1_3"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:101_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E28:101"><i><font color="Green" title="E97">A18</font></i></a>, <a class="txt" href="chord.html#E31:101"><i><font color="Green" title="E98">A19</font></i></a>, <a class="txt" href="chord.html#E1:101_3_1"><i><font color="Green" title="E101">A22</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2"><b>suppose </b></a><a NAME="E1:101_3_2"/><i><font color="Green" title="E101">A26</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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="c4">n</font> &amp; <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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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> <font color="Maroon" title="c3">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1"><b>suppose </b></a><a NAME="E1:101_3_2_1_1"/><i><font color="Green" title="E102">A27</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_3_2_1_1"/><b>then </b><i><font color="Green" title="E103">A28</font></i>: 
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">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="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E28:101"><i><font color="Green" title="E97">A18</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_2_1_1_1"/>
( <font color="Maroon" title="c9">k</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="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <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> <font color="Maroon" title="c3">m</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:101_3_2_1_1_1_1"/>
<font color="Maroon" title="c9">k</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="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E10:101"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:101_3_2_1_1_1_2"/><i><font color="Green" title="E104">A29</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_3_2_1_1_1_2"/>
( <font color="Maroon" title="c2">P</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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> )
 ;<br/><a NAME="E3:101_3_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="txt" href="chord.html#E2:101_3_2_1_1"><i><font color="Green" title="E103">A28</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E4:101_3_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:101"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="chord.html#E12:101"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="chord.html#E2:101_3_2_1_1"><i><font color="Green" title="E103">A28</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2_1_1_1_2"><i><font color="Green" title="E104">A29</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:101_3_2_1_1_1_3"/>
<font color="Maroon" title="c9">k</font> <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> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_3_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E104">A30</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E33:101"><i><font color="Green" title="E100">A21</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_3_2_1_1_1_3"/><i><font color="Green" title="E105">A31</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="chord.html#E1:101_3_2_1_1"><i><font color="Green" title="E102">A27</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E4:101_3_2_1_1_1_3"/>
( <font color="Maroon" title="c2">P</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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> )
 ;<br/><a NAME="E5:101_3_2_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="txt" href="chord.html#E2:101_3_2_1_1_1_3"><i><font color="Green" title="E104">A30</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E6:101_3_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E3:101"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="chord.html#E5:101"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="chord.html#E12:101"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="chord.html#E2:101_3_2_1_1_1_3"><i><font color="Green" title="E104">A30</font></i></a>, <a class="txt" href="chord.html#E3:101_3_2_1_1_1_3"><i><font color="Green" title="E105">A31</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2"><b>suppose </b></a><a NAME="E1:101_3_2_1_2"/><i><font color="Green" title="E102">A32</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_2_1_2_1"/>
( <font color="Maroon" title="c9">k</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="c3">m</font> or <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c9">k</font> <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> <font color="Maroon" title="c3">m</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:101_3_2_1_2_1_1"/>
<font color="Maroon" title="c9">k</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="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E7:101"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2_1_2"><i><font color="Green" title="E102">A32</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:101_3_2_1_2_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E7:101"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2_1_2"><i><font color="Green" title="E102">A32</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2_1_3"><b>suppose </b></a><a NAME="E1:101_3_2_1_2_1_3"/>
<font color="Maroon" title="c9">k</font> <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> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:101_3_2_1_2_1_3"/><b>then </b><i><font color="Green" title="E103">A33</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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="c2">P</font> )
 <b>by </b><i><a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E33:101"><i><font color="Green" title="E100">A21</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E3:101_3_2_1_2_1_3"/>
( <font color="Maroon" title="c2">P</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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> )
 ;<br/><a NAME="E4:101_3_2_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 <b>by </b><i><a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="txt" href="chord.html#E2:101_3_2_1_2_1_3"><i><font color="Green" title="E103">A33</font></i></a>, <a class="ref" href="glib_001.html#D24" title="GLIB_001:def.24">GLIB_001:def 24</a></i>;<br/><b>hence </b><a NAME="E5:101_3_2_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E5:101"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="chord.html#E12:101"><i><font color="Green" title="E89">A8</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2_1_2"><i><font color="Green" title="E102">A32</font></i></a>, <a class="txt" href="chord.html#E2:101_3_2_1_2_1_3"><i><font color="Green" title="E103">A33</font></i></a>, <a class="ref" href="glib_001.html#D31" title="GLIB_001:def.31">GLIB_001:def 31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_3"><b>suppose </b></a><a NAME="E1:101_3_2_1_3"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="chord.html#E6:101"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="chord.html#E28:101"><i><font color="Green" title="E97">A18</font></i></a>, <a class="txt" href="chord.html#E31:101"><i><font color="Green" title="E98">A19</font></i></a>, <a class="txt" href="chord.html#E1:101_3_2"><i><font color="Green" title="E101">A26</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
