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<div class="add">

<b>set </b><font color="Maroon" title="c5">W2</font> = <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>;<br/>
<a NAME="E1:127_1_1"/><i><font color="Green" title="E99">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <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">W</font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E63"><i><font color="Green" title="E55">Lm26</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_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="E1:127_1_1_1_1"/>
( ( not <font color="Maroon" title="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; not <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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">W</font> &amp; <font color="Maroon" title="c2">W</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">W</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="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <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">W</font> or  not <font color="Maroon" title="c2">W</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">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1"/><i><font color="Green" title="E100">A2</font></i>: 
( not <font color="Maroon" title="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; not <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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">W</font> &amp; <font color="Maroon" title="c2">W</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">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">m'</font> = <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c7">n'</font> = <font color="Maroon" title="c4">n</font> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:127_1_1_1_1_1_1"/><i><font color="Green" title="E101">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">b</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b</font> )
 ;<br/><a NAME="E3:127_1_1_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c8">a</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:127_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E102">A4</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_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="E1:127_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font> or ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E4:127_1_1_1_1_1_1"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green" title="E63">Lm34</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E103">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E66"><i><font color="Green" title="E58">Lm29</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_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="E1:127_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font> or ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">b</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E4:127_1_1_1_1_1_1"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green" title="E63">Lm34</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E66"><i><font color="Green" title="E58">Lm29</font></i></a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">b</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">W</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</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="E4:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E103">A5</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E104">A6</font></i></a>, <a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E104">A7</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">b</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">b2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n'</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E105">A8</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b2</font> &amp; <font color="Maroon" title="c10">b2</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">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E104">A7</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">b2</font> or <font color="Maroon" title="c10">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E103">A5</font></i></a>, <a class="txt" href="glib_001.html#E3:127_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E105">A8</font></i></a>, <a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E106">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E107">A10</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E106">A9</font></i></a></i>;<br/><a NAME="E5:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E6:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E4:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E107">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:127_1_1_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:127_1_1_1_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E103">A11</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">a2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n'</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">nm4</font> = <font color="Maroon" title="c7">n'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m'</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E104">A12</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a2</font> &amp; <font color="Maroon" title="c10">a2</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">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E103">A11</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font> or ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">b</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E4:127_1_1_1_1_1_1"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green" title="E63">Lm34</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E103">A11</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E105">A13</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c9">b</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">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c12">b2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n'</font> as  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E106">A14</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b2</font> &amp; <font color="Maroon" title="c12">b2</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">W</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E105">A13</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c10">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">b2</font> or <font color="Maroon" title="c12">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">a2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E107">A15</font></i>: 
<font color="Maroon" title="c10">a2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">b2</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m'</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E103">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">nm4</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="glib_001.html#E4:127_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E104">A12</font></i></a>, <a class="txt" href="glib_001.html#E3:127_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E106">A14</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><i><font color="Green" title="E107">A15</font></i></a>, <a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">a2</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">b</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">nm4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c11">nm4</font>
 ;<br/><b>hence </b><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:127_1_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:127_1_1_1_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:127_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:127_1_1_1_1_1"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_2"/>
( <font color="Maroon" title="c3">m</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or <font color="Maroon" title="c4">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> or  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <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">W</font> or  not <font color="Maroon" title="c2">W</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">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:127_1_1_1_1_2"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:127_1_1"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K9" title="GLIB_001:func.9">.remove</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a>
 ;<br/>


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