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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 not <font color="Olive" title="b1">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Olive" title="b1">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> holds <br/>for <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 <font color="Olive" title="b2">m</font> <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> holds <br/>(  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> iff <font color="Olive" title="b2">m</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="Olive" title="b3">n</font> )</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"> ( not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c2">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> implies for <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 <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> holds <br/>(  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> iff <font color="Olive" title="b1">m</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="Olive" title="b2">n</font> ) )</span><br/>



<b>assume </b><a NAME="E1:109"/><i><font color="Green" title="E87">A1</font></i>: 
( not <font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> &amp; not <font color="Maroon" title="c2">P</font> is <a href="chord.html#V5" title="CHORD:attr.5">chordal</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">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 <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> holds <br/>(  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> iff <font color="Olive" title="b1">m</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="Olive" title="b2">n</font> )</span><br/>

<b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <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="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> implies (  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> iff <font color="Maroon" title="c3">m</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> ) )</span><br/>

<b>assume </b><a NAME="E2:109"/><i><font color="Green" title="E88">A2</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> &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> )
 ; <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> iff <font color="Maroon" title="c3">m</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> )</span><br/>

<a NAME="E3:109"/><i><font color="Green" title="E89">A3</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="E4:109"/><b>then </b><i><font color="Green" title="E90">A4</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#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>
 
<b>by </b><i><a class="txt" href="#E1:109"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="#E2:109"><i><font color="Green" title="E88">A2</font></i></a>, <a class="ref" href="glib_001.html#T148" title="GLIB_001:th.148">GLIB_001:148</a></i>;<br/>
<a NAME="E5:109"/><i><font color="Green" title="E91">A5</font></i>: 
<font color="Maroon" title="c2">P</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>
 
<b>by </b><i><a class="txt" href="#E1:109"><i><font color="Green" title="E87">A1</font></i></a>, <a class="ref" href="chord.html#T32" target="_self" title="CHORD:th.32">Th32</a></i>;<br/>
<a NAME="E6:109"/><i><font color="Green" title="E92">A6</font></i>: 
<font color="Maroon" title="c3">m</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="#E2:109"><i><font color="Green" title="E88">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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"> ( <font color="Maroon" title="c3">m</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> implies  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> )</span>
<div class="add"><b>assume </b><a NAME="E1:109_1"/><i><font color="Green" title="E93">A7</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>
 ; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font></span><br/><a NAME="E2:109_1"/><i><font color="Green" title="E94">A8</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#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="#E2:109"><i><font color="Green" title="E88">A2</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="hs2(this)" href="javascript:()" title="109_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_1"/><i><font color="Green" title="E95">A9</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1"><b>now </b></a><div class="add"><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><a NAME="E1:109_1_1_1"/><i><font color="Green" title="E96">A10</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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>consider </b><font color="Maroon" title="c6">k</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:109_1_1_1"/><i><font color="Green" title="E97">A11</font></i>: 
( <font color="Maroon" title="c6">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> &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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E1:109_1_1_1"><i><font color="Green" title="E96">A10</font></i></a>, <a class="ref" href="glib_001.html#T101" title="GLIB_001:th.101">GLIB_001:101</a></i>;<br/><a NAME="E4:109_1_1_1"/><i><font color="Green" title="E98">A12</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="#E3:109_1_1_1"><i><font color="Green" title="E97">A11</font></i></a>, <a class="ref" href="chord.html#T4" target="_self" title="CHORD:th.4">Th4</a></i>;<br/><a NAME="E5:109_1_1_1"/><i><font color="Green" title="E99">A13</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="c6">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="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E3:109_1_1_1"><i><font color="Green" title="E97">A11</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><b>assume </b><a NAME="E6:109_1_1_1"/><i><font color="Green" title="E100">A14</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1_1_1_1"/>
( ( <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="c6">k</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="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> ) or ( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <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 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">k</font> ) )
 </span><b>by </b><i><a class="txt" href="#E5:109_1_1_1"><i><font color="Green" title="E99">A13</font></i></a>, <a class="txt" href="#E6:109_1_1_1"><i><font color="Green" title="E100">A14</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="109_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:109_1_1_1_1_1"/>
( <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="c6">k</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="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 ; <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="E2:109_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:109"><i><font color="Green" title="E88">A2</font></i></a>, <a class="txt" href="#E3:109"><i><font color="Green" title="E89">A3</font></i></a>, <a class="txt" href="#E5:109"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="#E6:109"><i><font color="Green" title="E92">A6</font></i></a>, <a class="txt" href="#E3:109_1_1_1"><i><font color="Green" title="E97">A11</font></i></a>, <a class="txt" href="#E4:109_1_1_1"><i><font color="Green" title="E98">A12</font></i></a>, <a class="ref" href="glib_001.html#D29" title="GLIB_001:def.29">GLIB_001:def 29</a></i>;<br/><b>hence </b><a NAME="E3:109_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:109_1_1"><i><font color="Green" title="E95">A9</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="109_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:109_1_1_1_1_2"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <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 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">k</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="E2:109_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="txt" href="#E2:109"><i><font color="Green" title="E88">A2</font></i></a>, <a class="txt" href="#E3:109"><i><font color="Green" title="E89">A3</font></i></a>, <a class="txt" href="#E5:109"><i><font color="Green" title="E91">A5</font></i></a>, <a class="txt" href="#E6:109"><i><font color="Green" title="E92">A6</font></i></a>, <a class="txt" href="#E3:109_1_1_1"><i><font color="Green" title="E97">A11</font></i></a>, <a class="txt" href="#E4:109_1_1_1"><i><font color="Green" title="E98">A12</font></i></a>, <a class="ref" href="glib_001.html#D29" title="GLIB_001:def.29">GLIB_001:def 29</a></i>;<br/><a NAME="E3:109_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E101">A15</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E4:109_1_1_1_1_2"/>
<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#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E5:109_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:109_1_1"><i><font color="Green" title="E95">A9</font></i></a>, <a class="txt" href="#E3:109_1_1_1_1_2"><i><font color="Green" title="E101">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><b>hence </b><a NAME="E3:109_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:109"><i><font color="Green" title="E87">A1</font></i></a>, <a class="txt" href="#E2:109"><i><font color="Green" title="E88">A2</font></i></a>, <a class="txt" href="#E4:109"><i><font color="Green" title="E90">A4</font></i></a>, <a class="txt" href="#E1:109_1"><i><font color="Green" title="E93">A7</font></i></a>, <a class="txt" href="#E1:109_1_1"><i><font color="Green" title="E95">A9</font></i></a>, <a class="ref" href="chord.html#D10" target="_self" title="CHORD:def.10">Def10</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>hence </b><a NAME="E4:109_1"/>
<font color="Maroon" title="c3">m</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>
 <b>by </b><i><a class="txt" href="#E2:109_1"><i><font color="Green" title="E94">A8</font></i></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>

<b>assume </b><a NAME="E8:109"/><i><font color="Green" title="E93">A16</font></i>: 
<font color="Maroon" title="c3">m</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">  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></span><br/>

<b>take </b>
<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="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
; <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</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="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="E9:109"/>
<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="#E2:109"><i><font color="Green" title="E88">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="E10:109"/>
<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="c3">m</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="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="#E3:109"><i><font color="Green" title="E89">A3</font></i></a>, <a class="txt" href="#E8:109"><i><font color="Green" title="E93">A16</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</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>
