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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>2</sub></font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> &amp;  ex <font color="Olive">Wn</font> being   <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">Wn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Wn</font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Olive">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) or ( <font color="Maroon">a<sub>2</sub></font> is  not    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> ) );<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_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="E1:134_1_1_1_1_1"/>
( <font color="Maroon">x</font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> or  not <font color="Maroon">x</font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_1_1_1_1"/>
<font color="Maroon">x</font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">x</font> as    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font> ;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">W</font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E3:134_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">W</font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>]
 ;<br/><b>hence </b><a NAME="E4:134_1_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_1_1_1_2"/>
<font color="Maroon">x</font> is  not    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:134_1_1_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:134_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:134_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:134_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1">RECDEF_1:sch 1</a>(, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font>;<br/>
<a NAME="E5:134_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:134_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font>
 ;<br/><b>then consider </b><font color="Maroon">Wn</font> being    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:134_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">Wn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:134_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> 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>
<a NAME="E7:134_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E8:134_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>then reconsider </b><font color="Maroon">W0</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 as    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">Wn</font> being   <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font> st <br/>( <font color="Olive">Wn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Wn</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; ( for <font color="Olive">m</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</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">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">Wn</font> holds <br/><font color="Olive">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> ) );<br/>
<a NAME="E10:134_1_1"/>
for <font color="Olive">m</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</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> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W0</font> holds <br/><font color="Maroon">W0</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<a NAME="E11:134_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:134_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>then consider </b><font color="Maroon">Wn</font> being    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:134_1_1_3"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">Wn</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; ( for <font color="Olive">m</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> holds <br/><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> ) )
 ;<br/><b>set </b><font color="Maroon">a</font> = <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">b</font> = <font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">Wn1</font> = <font color="Maroon">Wn</font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">Wn1</font> = <font color="Maroon">Wn</font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">Wn1</font> = <font color="Maroon">Wn1</font>;<br/><a NAME="E5:134_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:134_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn1</font>
 <b>by </b><i/>;<br/><a NAME="E7:134_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:134_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn1</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/><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>set </b><font color="Maroon">m'</font> = <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font>;<br/><b>assume </b><a NAME="E9:134_1_1_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">m</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn1</font> )
 ;<br/><a NAME="E10:134_1_1_3"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:134_1_1_3"/><b>then </b>
<font color="Maroon">m</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E12:134_1_1_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <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="E13:134_1_1_3"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font>
 <b>by </b><i/>;<br/><a NAME="E14:134_1_1_3"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">W1</font> = <font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_3_1_1"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <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">Wn</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <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">Wn</font>
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> &amp; <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> &amp; <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>by </b><i/>;<br/><a NAME="E3:134_1_1_3_1_1_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> &amp; <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> &amp; <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i>, </i>;<br/><a NAME="E4:134_1_1_3_1_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E5:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">Wn1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1_3_1_1_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:134_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:134_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, </i>;<br/><b>hence </b><a NAME="E4:134_1_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, </i>;<br/></div><b>end;</b></div><a NAME="E8:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:134_1_1_3_1_1_1"/><i><font color="Green">E96</font></i>: 
( not 1 is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><i><font color="Green">E97</font></i>: <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  = 
<font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font></span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a> 
<b>by </b><i>, <a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E49</font></i></a></i>
;<br/><b>set </b><font color="Maroon">m'</font> = <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font>;<br/><a NAME="E11:134_1_1_3_1_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E12:134_1_1_3_1_1_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn1</font> &amp; <font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">k</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn1</font> &amp; <font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> ) )
 <b>by </b><i>, </i>;<br/><a NAME="E13:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D14" target="_self">Def14</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_3_1_1_1_3_1"/>
( <font color="Maroon">m</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">m</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:134_1_1_3_1_1_1_3_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, , </i>;<br/><b>then </b><i><font color="Green">E106</font></i>: <font color="Maroon">Wn1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font>
<b>by </b><i>, , <a class="ref" href="glib_001.html#D19" target="_self">Def19</a>, </i>
<br/>.= 
<font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, <a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E53</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">maa1</font> = <font color="Maroon">m</font> <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/><a NAME="E6:134_1_1_3_1_1_1_3_1_1"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">maa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E7:134_1_1_3_1_1_1_3_1_1"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">maa1</font> <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="E8:134_1_1_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">maa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:134_1_1_3_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, , , <a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E10:134_1_1_3_1_1_1_3_1_1"/>
<font color="Maroon">maa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, </i>;<br/><a NAME="E11:134_1_1_3_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E256</font></i>: 
<font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_3_1_1_1_3_1_1_2_1"/>
( <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E258</font></i>: 
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:134_1_1_3_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E259</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then </b><i><font color="Green">E260</font></i>: <font color="Maroon">Wn1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D19" target="_self">Def19</a>, <a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, </i>
<br/>.= 
<font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E53</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">m'aa1</font> = <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></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="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">m'aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <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="E7:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">m'aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D21" target="_self">Def21</a>, </i>;<br/><a NAME="E8:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E263</font></i>: 
<font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m'aa1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D21" target="_self">Def21</a>, <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E9:134_1_1_3_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon">m'aa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, </i>;<br/><a NAME="E10:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D21" target="_self">Def21</a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, , <a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E11:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, , , <a class="ref" href="glib_001.html#D22" target="_self">Def22</a></i>;<br/><b>hence </b><a NAME="E12:134_1_1_3_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_1_2_1_2"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:134_1_1_3_1_1_1_3_1_1_2_1_2"/><i><font color="Green">E265</font></i>: 
( <font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</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"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> 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"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</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#T6" target="_self">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="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#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:134_1_1_3_1_1_1_3_1_1_2_1_2"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, , , <a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, <a class="ref" href="glib_001.html#D21" target="_self">Def21</a>, <a class="ref" href="glib_001.html#D28" target="_self">Def28</a></i>;<br/><a NAME="E5:134_1_1_3_1_1_1_3_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:134_1_1_3_1_1_1_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:134_1_1_3_1_1_1_3_1_1_2_1_2"/>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:134_1_1_3_1_1_1_3_1_1"/>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_2"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">m</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">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, </i>;<br/><b>then </b><i><font color="Green">E268</font></i>: <font color="Maroon">Wn1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font>
<b>by </b><i>, , <a class="ref" href="glib_001.html#D19" target="_self">Def19</a>, </i>
<br/>.= 
<font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, <a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E53</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">maa1</font> = <font color="Maroon">m</font> <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/><a NAME="E5:134_1_1_3_1_1_1_3_1_2"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">maa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_2"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">maa1</font> <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="E7:134_1_1_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">maa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:134_1_1_3_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E273</font></i>: 
<font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, , , <a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E9:134_1_1_3_1_1_1_3_1_2"/>
<font color="Maroon">maa1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, </i>;<br/><a NAME="E10:134_1_1_3_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</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="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E48</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1_3_1_1_1_3_1_2_2"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E276</font></i>: 
<font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, </i>;<br/><b>set </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E3:134_1_1_3_1_1_1_3_1_2_2"/>
( <font color="Maroon">Wn1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</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"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> 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"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</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#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, , , <a class="ref" href="glib_001.html#D21" target="_self">Def21</a></i>;<br/><a NAME="E5:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E7:134_1_1_3_1_1_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:134_1_1_3_1_1_1_3_1_2"/>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:134_1_1_3_1_1_1"/>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_2"/><i><font color="Green">E277</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">Wn</font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> &amp; <font color="Maroon">Wn</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Wn</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:134_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E278</font></i>: 
<font color="Maroon">Wn1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Wn</font>
 <b>by </b><i><a class="ref" href="glib_001.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:134_1_1_3_1_1_2"/>
<font color="Maroon">m</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">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:134_1_1_3_1_1_2"/>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <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="E16:134_1_1_3"/>
<font color="Maroon">Wn1</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E13:134_1_1"/><b>then </b><i><font color="Green">E279</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E14:134_1_1"/><i><font color="Green">E280</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>set </b><font color="Maroon">n</font> = <span class="p1">(<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> 1;<br/>
<b>consider </b><font color="Maroon">P</font> being    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E16:134_1_1"/><i><font color="Green">E281</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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> 1</span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; ( for <font color="Olive">m</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</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> <span class="p2">(<span class="default"><span class="p3">(<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> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> holds <br/><font color="Maroon">P</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:134_1_1_4"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:134_1_1_4"/><i><font color="Green">E283</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<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> 1
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:134_1_1_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <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> 1</span>)</span> <a href="nat_1.html#K1">+</a> <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> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:134_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<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> 1</span>)</span></span>)</span> <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="E5:134_1_1_4"/><b>then </b>
<font color="Maroon">m</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> <span class="p2">(<span class="default"><span class="p3">(<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> 1</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>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:134_1_1_4"/>
<font color="Maroon">P</font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_001.html#D14" target="_self">Def14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:134_1_1"/><b>then </b>
<font color="Maroon">P</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E19:134_1_1"/>
( <font color="Maroon">P</font> is <a href="glib_001.html#V4" target="_self">Trail-like</a> &amp; <font color="Maroon">P</font> is <a href="glib_001.html#V5" target="_self">Path-like</a> )
 <b>by </b><i/>;<br/>


</div>
