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

<span class="kw">set </span><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"/><span class="lab"><font color="Green" title="E99">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E63"><span class="lab"><font color="Green" title="E55">Lm26</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:127_1_1_1_1_1"/><span class="lab"><font color="Green" title="E100">A2</font></span>: 
( 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">m9</font> = <font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c7">n9</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"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><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> <span class="kw"> such that </span><a NAME="E2:127_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E101">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">and </span><br/><a NAME="E3:127_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E102">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:127_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E103">A5</font></span>: 
<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 class="kw">and </span><br/><a NAME="E5:127_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E104">A6</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:127_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="txt" href="#E3:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:127_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A7</font></span>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E103">A5</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:127_1_1_1_1_1_1"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="txt" href="#E4:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E103">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:127_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A8</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E9:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E106">A8</font></span></a>, <a class="txt" href="glib_001.html#E71"><span class="lab"><font color="Green" title="E63">Lm34</font></span></a></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"><span class="kw">suppose </span></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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="glib_001.html#E66"><span class="lab"><font color="Green" title="E58">Lm29</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E7:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A7</font></span></a>, <a class="txt" href="glib_001.html#E71"><span class="lab"><font color="Green" title="E63">Lm34</font></span></a></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_1"><span class="kw">suppose </span></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E108">A10</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E109">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1"><span class="lab"><font color="Green" title="E99">A1</font></span></a>, <a class="txt" href="#E4:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E103">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E108">A10</font></span></a>, <a class="txt" href="glib_001.html#E66"><span class="lab"><font color="Green" title="E58">Lm29</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="txt" href="#E3:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="txt" href="#E5:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A6</font></span></a>, <a class="txt" href="#E2:127_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E107">A9</font></span></a>, <a class="txt" href="#E2:127_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E109">A11</font></span></a>, <a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E108">A12</font></span>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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">m9</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E109">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E108">A12</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E110">A14</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E108">A12</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E101">A3</font></span></a>, <a class="txt" href="#E5:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A6</font></span></a>, <a class="txt" href="#E2:127_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E107">A9</font></span></a>, <a class="txt" href="#E4:127_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E110">A14</font></span></a>, <a class="txt" href="#E3:127_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E109">A13</font></span></a>, <a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E111">A15</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E112">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E113">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></span>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E111">A15</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E5:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:127_1_1_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:127_1_1_1_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E107">A18</font></span>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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">m9</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">nm4</font> = <font color="Maroon" title="c7">n9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E108">A19</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E107">A18</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E7:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A7</font></span></a>, <a class="txt" href="glib_001.html#E71"><span class="lab"><font color="Green" title="E63">Lm34</font></span></a></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_1"><span class="kw">suppose </span></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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E107">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E109">A20</font></span>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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">m9</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n9</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E110">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E111">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1"><span class="lab"><font color="Green" title="E100">A2</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A20</font></span></a>, <a class="txt" href="glib_001.html#E67"><span class="lab"><font color="Green" title="E59">Lm30</font></span></a></span>;<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"><span class="kw">now </span></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"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E112">A23</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><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">m9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E107">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">nm4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A6</font></span></a>, <a class="txt" href="#E4:127_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E108">A19</font></span></a>, <a class="txt" href="#E4:127_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E111">A22</font></span></a>, <a class="txt" href="#E3:127_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E110">A21</font></span></a>, <a class="txt" href="#E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E112">A23</font></span></a>, <a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">b</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">nm4</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E102">A4</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:127_1_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:127_1_1_1_1_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:127_1_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E94"><span class="lab"><font color="Green" title="E86">Lm57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_2"><span class="kw">suppose </span></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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><div class="add"><span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
