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

<b>set </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">es1</font> = <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_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="E1:133_1_1_1_1"/>
( ( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> ) or <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> or  not <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font>, <font color="Maroon">n'</font> = <font color="Maroon">n</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:133_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i/>;<br/><a NAME="E4:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">W</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i/>;<br/><a NAME="E5:133_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">maa1</font> = <font color="Maroon">m'</font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">lenWn4</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n'</font> as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E9:133_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">lenWaa1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E11:133_1_1_1_1_1"/>
( 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">maa1</font> &amp; 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">lenWn4</font> &amp; 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">lenWaa1</font> &amp; 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="pepin.html#T22">PEPIN:22</a></i>;<br/><a NAME="E12:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">maa1</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> &amp; <font color="Maroon">lenWn4</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> &amp; <font color="Maroon">lenWaa1</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> ;<br/><b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> ;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> ;<br/><b>set </b><font color="Maroon">es</font> = <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> </span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:133_1_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_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="E1:133_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:133_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 )
 ;<br/><a NAME="E4:133_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:133_1_1_1_1_1_1_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:133_1_1_1_1_1_1_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E7:133_1_1_1_1_1_1_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:133_1_1_1_1_1_1_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:133_1_1_1_1_1_1_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E10:133_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:133_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 )
 ;<br/><a NAME="E4:133_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:133_1_1_1_1_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:133_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:133_1_1_1_1_1_1_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E8:133_1_1_1_1_1_1_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:133_1_1_1_1_1_1_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E10:133_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:133_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 ;<br/></div><b>end;</b></div><a NAME="E14:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E15:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E16:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E17:133_1_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span> &amp; <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T11">XBOOLE_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> , <font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:133_1_1_1_1_1_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/><b>then consider </b><font color="Maroon">a'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:133_1_1_1_1_1_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">a'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 )
 ;<br/><b>consider </b><font color="Maroon">b'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:133_1_1_1_1_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">b'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 )
 <b>by </b><i><a class="ref" href="glib_001.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E6:133_1_1_1_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:133_1_1_1_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:133_1_1_1_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:133_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E10:133_1_1_1_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E11:133_1_1_1_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:133_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b'</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:133_1_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E22:133_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  is    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#T28">GRAPH_2:28</a></i>;<br/><b>then reconsider </b><font color="Maroon">es</font> = <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 ) </span>}</span> </span>)</span> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#T29">GRAPH_2:29</a></i>;<br/><a NAME="E24:133_1_1_1_1_1"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E25:133_1_1_1_1_1"/><i><font color="Green">E96</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">es2</font> =  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font>;<br/><a NAME="E26:133_1_1_1_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E27:133_1_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a>, </i>;<br/><a NAME="E28:133_1_1_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>then </b><i><font color="Green">E100</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>
;<br/><a NAME="E30:133_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#D14" target="_self">Def14</a>, , </i>;<br/><a NAME="E31:133_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E32:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 ;<br/><div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:133_1_1_1_1_1_4_1"/>
( <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_4_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_4_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:133_1_1_1_1_1_4_1_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:133_1_1_1_1_1_4_1_1_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:133_1_1_1_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:133_1_1_1_1_1_4_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i>, <a class="ref" href="card_2.html#T19">CARD_2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_4_1_2"/>
<font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:133_1_1_1_1_1_4_1_2"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:133_1_1_1_1_1_4_1_2"/>
<font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:133_1_1_1_1_1_4_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="ref" href="graph_2.html#T4">GRAPH_2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><i><font color="Green">E254</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:133_1_1_1_1_1_5_1"/>
( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 or <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_5_1_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2
 ;<br/><div class="add"><div><i><font color="Green">E256</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_5_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:133_1_1_1_1_1_5_1_1_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:133_1_1_1_1_1_5_1_1_1"/><i><font color="Green">E258</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 )
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a></i>;<br/><b>thus </b><a NAME="E6:133_1_1_1_1_1_5_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E3:133_1_1_1_1_1_5_1_1"/>
<font color="Maroon">lenWaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E4:133_1_1_1_1_1_5_1_1"/><b>then </b>
<font color="Maroon">lenWaa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:133_1_1_1_1_1_5_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E6:133_1_1_1_1_1_5_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:133_1_1_1_1_1_5_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:133_1_1_1_1_1_5_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:133_1_1_1_1_1_5_1_1"/><b>then </b>
<font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_2.html#T4">NAT_2:4</a></i>;<br/><b>hence </b><a NAME="E10:133_1_1_1_1_1_5_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i>, <a class="ref" href="card_2.html#T19">CARD_2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_5_1_2"/>
<font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_5_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> ) </span>}</span> 
 ;<br/><a NAME="E4:133_1_1_1_1_1_5_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T4">GRAPH_2:4</a></i>;<br/><a NAME="E5:133_1_1_1_1_1_5_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lenWn4</font>
 <b>by </b><i/>;<br/><a NAME="E6:133_1_1_1_1_1_5_1_2"/><b>then </b>
( <font color="Maroon">lenWn4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 ;<br/><b>hence </b><a NAME="E7:133_1_1_1_1_1_5_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_6"/>
not <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:133_1_1_1_1_1_6"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:133_1_1_1_1_1_6"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:133_1_1_1_1_1_6"/><i><font color="Green">E261</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2 )
 ;<br/><b>consider </b><font color="Maroon">y'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:133_1_1_1_1_1_6"/><i><font color="Green">E263</font></i>: 
( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 )
 <b>by </b><i/>;<br/><a NAME="E9:133_1_1_1_1_1_6"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:133_1_1_1_1_1_6"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:133_1_1_1_1_1_6"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:133_1_1_1_1_1_6"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">y'</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E13:133_1_1_1_1_1_6"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/></div><b>end;</b></div><a NAME="E36:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E264</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E37:133_1_1_1_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E38:133_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">Z</font></span>)</span> <a href="nat_1.html#K2">*</a> 2 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E39:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E265</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E40:133_1_1_1_1_1"/><i><font color="Green">E266</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/><a NAME="E41:133_1_1_1_1_1"/><i><font color="Green">E267</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T45">FINSEQ_3:45</a></i>;<br/><a NAME="E42:133_1_1_1_1_1"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_7"/>
<font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:133_1_1_1_1_1_7"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E4:133_1_1_1_1_1_7"/><b>then </b>
<font color="Maroon">n'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><b>hence </b><a NAME="E5:133_1_1_1_1_1_7"/>
contradiction
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">n1div2aa1</font> = <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>set </b><font color="Maroon">lenY</font> = <span class="p1">(<span class="default"><font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1div2aa1</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_8"/>
<font color="Maroon">n1div2aa1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2
 ;<br/><a NAME="E2:133_1_1_1_1_1_8"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n1div2aa1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_8"/><b>then </b>
<font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">lenWaa1</font>
 <b>by </b><i/>;<br/><a NAME="E4:133_1_1_1_1_1_8"/><b>then </b><i><font color="Green">E271</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E5:133_1_1_1_1_1_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E6:133_1_1_1_1_1_8"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">lenY</font> = <span class="p1">(<span class="default"><font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n1div2aa1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E47:133_1_1_1_1_1"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lenY</font> ) </span>}</span> 
 ;<br/><a NAME="E48:133_1_1_1_1_1"/><i><font color="Green">E273</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E49:133_1_1_1_1_1"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">lenY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenWn4</font>
 <b>by </b><i/>;<br/><a NAME="E50:133_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">lenWn4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">lenY</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 ;<br/><a NAME="E51:133_1_1_1_1_1"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">lenY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:133_1_1_1_1_1_9"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:133_1_1_1_1_1_9"/><i><font color="Green">E276</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:133_1_1_1_1_1_9"/><b>then </b><i><font color="Green">E277</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:133_1_1_1_1_1_9_1_1"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> or <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_9_1_1_1"/><i><font color="Green">E278</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:133_1_1_1_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E279</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_9_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E4:133_1_1_1_1_1_9_1_1_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:133_1_1_1_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E280</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="glib_001.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E6:133_1_1_1_1_1_9_1_1_1"/><i><font color="Green">E281</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, , <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E7:133_1_1_1_1_1_9_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E8:133_1_1_1_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E282</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E9:133_1_1_1_1_1_9_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, <a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="glib_001.html#D19" target="_self">Def19</a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a>, <a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>
;<br/><a NAME="E11:133_1_1_1_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E283</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E12:133_1_1_1_1_1_9_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>;<br/><a NAME="E13:133_1_1_1_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E284</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:133_1_1_1_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E285</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E15:133_1_1_1_1_1_9_1_1_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E16:133_1_1_1_1_1_9_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_1_9_1_1_2"/><i><font color="Green">E286</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:133_1_1_1_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_9_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E4:133_1_1_1_1_1_9_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:133_1_1_1_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E288</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:133_1_1_1_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E289</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E7:133_1_1_1_1_1_9_1_1_2"/><i><font color="Green">E290</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, , <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><div><i><font color="Green">E291</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_9_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_9_1_1_2_1"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2
 ;<br/><a NAME="E2:133_1_1_1_1_1_9_1_1_2_1"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_9_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">maa1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:133_1_1_1_1_1_9_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:133_1_1_1_1_1_9_1_1_2"/><i><font color="Green">E292</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E12:133_1_1_1_1_1_9_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="glib_001.html#D30" target="_self">Def30</a></i>;<br/><a NAME="E13:133_1_1_1_1_1_9_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E14:133_1_1_1_1_1_9_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E15:133_1_1_1_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E293</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E294</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_9_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_9_1_1_2_2"/>
<font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 ;<br/><a NAME="E2:133_1_1_1_1_1_9_1_1_2_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">lenWn4</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_9_1_1_2_2"/><b>then </b>
<font color="Maroon">lenWn4</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">maa1</font>
 <b>by </b><i/>;<br/><a NAME="E4:133_1_1_1_1_1_9_1_1_2_2"/><b>then </b><i><font color="Green">E295</font></i>: 
<font color="Maroon">lenWn4</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>set </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n'</font>;<br/><b>reconsider </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E6:133_1_1_1_1_1_9_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E7:133_1_1_1_1_1_9_1_1_2_2"/><b>then </b>
<font color="Maroon">z</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E8:133_1_1_1_1_1_9_1_1_2_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E17:133_1_1_1_1_1_9_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="glib_001.html#D21" target="_self">Def21</a>, <a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, <a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a></i>
;<br/><a NAME="E19:133_1_1_1_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E296</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E20:133_1_1_1_1_1_9_1_1_2"/><i><font color="Green">E297</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E21:133_1_1_1_1_1_9_1_1_2"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><i><font color="Green">E299</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1_1_9_1_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_1_1_1_9_1_1_2_4"/>
<font color="Maroon">lenWaa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1div2aa1</font>
 ;<br/><a NAME="E2:133_1_1_1_1_1_9_1_1_2_4"/><b>then </b>
<font color="Maroon">lenWaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n1div2aa1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E3:133_1_1_1_1_1_9_1_1_2_4"/><b>then </b>
<font color="Maroon">lenWaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">maa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n1div2aa1</font></span>)</span>
 ;<br/><a NAME="E4:133_1_1_1_1_1_9_1_1_2_4"/><b>then </b>
<font color="Maroon">lenWaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">maa1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n1div2aa1</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:133_1_1_1_1_1_9_1_1_2_4"/><b>then </b>
<font color="Maroon">lenWaa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">maa1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:133_1_1_1_1_1_9_1_1_2_4"/><b>then </b><i><font color="Green">E300</font></i>: 
<font color="Maroon">lenWaa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, , <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E8:133_1_1_1_1_1_9_1_1_2_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E9:133_1_1_1_1_1_9_1_1_2_4"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E23:133_1_1_1_1_1_9_1_1_2"/><b>then </b>
<font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a></i>;<br/><a NAME="E24:133_1_1_1_1_1_9_1_1_2"/><b>then </b>
<font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E25:133_1_1_1_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E301</font></i>: 
<font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>consider </b><font color="Maroon">lenWaa1'</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E27:133_1_1_1_1_1_9_1_1_2"/><i><font color="Green">E303</font></i>: 
( <font color="Maroon">lenWaa1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenWaa1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lenWaa1'</font> <a href="nat_1.html#K5">div</a> 2 )
 <b>by </b><i><a class="ref" href="glib_001.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E28:133_1_1_1_1_1_9_1_1_2"/><i><font color="Green">E304</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="txt" href="glib_001.html#E44"><i><font color="Green">Lemma147</font></i></a>, <a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a>, <a class="txt" href="glib_001.html#E46"><i><font color="Green">Lemma149</font></i></a>, </i>;<br/><a NAME="E29:133_1_1_1_1_1_9_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n1div2aa1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, </i>;<br/><b>hence </b><a NAME="E30:133_1_1_1_1_1_9_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, , <a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a>, <a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</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="E6:133_1_1_1_1_1_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 ;<br/></div><b>end;</b></div><a NAME="E53:133_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E54:133_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:133_1_1_1_1_2"/>
( <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> or  not <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:133_1_1_1_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:133_1_1_1_1_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:133_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font>
 ;<br/>


</div>
