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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( <font color="Maroon" title="c2">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> &amp;  ex <font color="Olive" title="b1">Wn</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">Wn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Wn</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ) ) or ( <font color="Maroon" title="c2">W</font> is  not    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> ) );<br/>
<div><i><font color="Green" title="E103">A1</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" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]</span><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" title="c4">x</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> or  not <font color="Maroon" title="c4">x</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">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" title="c4">x</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">W</font> = <font color="Maroon" title="c4">x</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font> ;<br/><b>set </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">W</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">W</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>]
 ;<br/><b>hence </b><a NAME="E4:134_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> is  not    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><b>hence </b><a NAME="E2:134_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:134_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:134_1_1"/><i><font color="Green" title="E104">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">y2</font>] holds <br/><font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:134_1_1"/><i><font color="Green" title="E105">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1" title="RECDEF_1:sch.1">RECDEF_1:sch 1</a>(<a class="txt" href="glib_001.html#E1:134_1_1"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1"><i><font color="Green" title="E104">A2</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">G</font> is    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">W0</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1"><i><font color="Green" title="E105">A3</font></i></a>, <a class="txt" href="glib_001.html#E108"><i><font color="Green" title="E100">Lm70</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> ex <font color="Olive" title="b1">Wn</font> being   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font> st <br/>( <font color="Olive" title="b1">Wn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Wn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b2">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Wn</font> holds <br/><font color="Olive" title="b1">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> ) );<br/>
<a NAME="E6:134_1_1"/>
for <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W0</font> holds <br/><font color="Maroon" title="c4">W0</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/>
<a NAME="E7:134_1_1"/><b>then </b><i><font color="Green" title="E106">A4</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1"><i><font color="Green" title="E105">A3</font></i></a></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" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">n</font>] implies  ex <font color="Olive" title="b1">Wn1</font> being   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Wn1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Wn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b2">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Wn1</font> holds <br/><font color="Olive" title="b1">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> ) ) )</span><br/><b>assume </b><a NAME="E1:134_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Wn1</font> being   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font> st <br/>( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Wn1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Wn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b2">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Wn1</font> holds <br/><font color="Olive" title="b1">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> ) )</span><br/><b>then consider </b><font color="Maroon" title="c6">Wn</font> being    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E3:134_1_1_2"/><i><font color="Green" title="E107">A5</font></i>: 
( <font color="Maroon" title="c6">Wn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> holds <br/><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> ) )
 ;<br/><b>set </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">Wn1</font> = <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c10">Wn1</font> = <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="txt" href="glib_001.html#E109"><i><font color="Green" title="E101">Lm71</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c11">Wn1</font> = <font color="Maroon" title="c10">Wn1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Wn1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> holds <br/><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> ) )</span><br/><a NAME="E5:134_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1"><i><font color="Green" title="E105">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:134_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Wn1</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_2"><i><font color="Green" title="E107">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> holds <br/><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> ) )</span><br/><a NAME="E7:134_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E63"><i><font color="Green" title="E55">Lm26</font></i></a></i>;<br/><b>hence </b><a NAME="E8:134_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_2"><i><font color="Green" title="E107">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> holds <br/><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font></span><br/><b>let </b><font color="Maroon" title="c12">m</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> implies <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font> )</span><br/><b>assume </b><a NAME="E9:134_1_1_2"/><i><font color="Green" title="E108">A6</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font></span><br/><a NAME="E10:134_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:134_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E12:134_1_1_2"/><b>then </b><i><font color="Green" title="E109">A7</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:134_1_1_2"/><i><font color="Green" title="E110">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E63"><i><font color="Green" title="E55">Lm26</font></i></a></i>;<br/><a NAME="E14:134_1_1_2"/><b>then </b><i><font color="Green" title="E111">A9</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c13">W1</font> = <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">W2</font> = <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_2_1_1"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_2_1_1_1"/><i><font color="Green" title="E112">A10</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font></span><br/><div class="add"><a NAME="E2:134_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E113">A11</font></i>: 
( <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> &amp; <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> &amp; <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self" title="GLIB_001:def.20">Def20</a></i>;<br/><a NAME="E3:134_1_1_2_1_1_1"/><i><font color="Green" title="E114">A12</font></i>: 
( <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> &amp; <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> &amp; <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1"><i><font color="Green" title="E112">A10</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self" title="GLIB_001:def.22">Def22</a></i>;<br/><a NAME="E4:134_1_1_2_1_1_1"/><i><font color="Green" title="E115">A13</font></i>: 
( <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1"><i><font color="Green" title="E112">A10</font></i></a>, <a class="txt" href="glib_001.html#E86"><i><font color="Green" title="E78">Lm49</font></i></a>, <a class="txt" href="glib_001.html#E87"><i><font color="Green" title="E79">Lm50</font></i></a></i>;<br/><a NAME="E5:134_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E116">A14</font></i>: 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:134_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E117">A15</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1"><i><font color="Green" title="E113">A11</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1"><i><font color="Green" title="E114">A12</font></i></a>, <a class="ref" href="glib_001.html#D12" target="_self" title="GLIB_001:def.12">Def12</a></i>;<br/><div><i><font color="Green" title="E118">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1_2_1_1_1_1"/><i><font color="Green" title="E119">A17</font></i>: 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:134_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1"><i><font color="Green" title="E112">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:134_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_2"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_1"><i><font color="Green" title="E119">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E4:134_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1"><i><font color="Green" title="E112">A10</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1"><i><font color="Green" title="E113">A11</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_1"><i><font color="Green" title="E119">A17</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self" title="GLIB_001:def.22">Def22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E8:134_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E119">A18</font></i>: 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_2_1_1_1"><i><font color="Green" title="E115">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:134_1_1_2_1_1_1"/><i><font color="Green" title="E120">A19</font></i>: 
( not 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a></i>;<br/><b>then </b><i><font color="Green" title="E121">A20</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  = 
<font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1"><i><font color="Green" title="E113">A11</font></i></a>, <a class="txt" href="glib_001.html#E53"><i><font color="Green" title="E45">Lm16</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font></span>)</span></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1"><i><font color="Green" title="E114">A12</font></i></a>, <a class="txt" href="glib_001.html#E53"><i><font color="Green" title="E45">Lm16</font></i></a></i>
;<br/><b>set </b><font color="Maroon" title="c15">m'</font> = <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font>;<br/><a NAME="E11:134_1_1_2_1_1_1"/><i><font color="Green" title="E122">A21</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E87"><i><font color="Green" title="E79">Lm50</font></i></a></i>;<br/><a NAME="E12:134_1_1_2_1_1_1"/><i><font color="Green" title="E123">A22</font></i>: 
( <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> &amp; <font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wn1</font> &amp; <font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self" title="GLIB_001:def.22">Def22</a></i>;<br/><a NAME="E13:134_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E124">A23</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E13:134_1_1_2"><i><font color="Green" title="E110">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_2_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_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_2_1_1_1_3_1"/>
( <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_2"><i><font color="Green" title="E109">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_2_1_1_1_3_1_1"/><i><font color="Green" title="E125">A24</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font></span><br/><div class="add"><a NAME="E2:134_1_1_2_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E126">A25</font></i>: 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_2"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="glib_001.html#E14:134_1_1_2"><i><font color="Green" title="E111">A9</font></i></a></i>;<br/><a NAME="E3:134_1_1_2_1_1_1_3_1_1"/>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1"><i><font color="Green" title="E112">A10</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_2_1_1_1"><i><font color="Green" title="E119">A18</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E125">A24</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E127">A26</font></i>: <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font>
<b>by </b><i><a class="txt" href="glib_001.html#E6:134_1_1_2_1_1_1"><i><font color="Green" title="E117">A15</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2_1_1_1"><i><font color="Green" title="E120">A19</font></i></a>, <a class="txt" href="glib_001.html#E10:134_1_1_2_1_1_1"><i><font color="Green" title="E121">A20</font></i></a>, <a class="txt" href="glib_001.html#E58"><i><font color="Green" title="E50">Lm21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font>
<b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_2"><i><font color="Green" title="E109">A7</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_2_1_1_1"><i><font color="Green" title="E119">A18</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c16">maa1</font> = <font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E6:134_1_1_2_1_1_1_3_1_1"/><i><font color="Green" title="E128">A27</font></i>: 
<font color="Maroon" title="c16">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 ;<br/><a NAME="E7:134_1_1_2_1_1_1_3_1_1"/><i><font color="Green" title="E129">A28</font></i>: 
<font color="Maroon" title="c16">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E8:134_1_1_2_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c16">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a></i>;<br/><a NAME="E9:134_1_1_2_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E130">A29</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2_1_1_1"><i><font color="Green" title="E120">A19</font></i></a>, <a class="txt" href="glib_001.html#E4:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E127">A26</font></i></a>, <a class="txt" href="glib_001.html#E6:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E128">A27</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green" title="E44">Lm15</font></i></a></i>;<br/><a NAME="E10:134_1_1_2_1_1_1_3_1_1"/>
<font color="Maroon" title="c16">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E13:134_1_1_2"><i><font color="Green" title="E110">A8</font></i></a>, <a class="txt" href="glib_001.html#E7:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E129">A28</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:134_1_1_2_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E131">A30</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_2"><i><font color="Green" title="E111">A9</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2_1_1_1"><i><font color="Green" title="E120">A19</font></i></a>, <a class="txt" href="glib_001.html#E6:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E128">A27</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E130">A29</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green" title="E44">Lm15</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_2_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_2_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_2_1_1_1_3_1_1_2_1"/>
( <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_1_1_1_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_2_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E132">A31</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font></span><br/><div class="add"><a NAME="E2:134_1_1_2_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E133">A32</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1"><i><font color="Green" title="E113">A11</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a></i>;<br/><a NAME="E3:134_1_1_2_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E134">A33</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><b>then </b><i><font color="Green" title="E135">A34</font></i>: <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E6:134_1_1_2_1_1_1"><i><font color="Green" title="E117">A15</font></i></a>, <a class="txt" href="glib_001.html#E10:134_1_1_2_1_1_1"><i><font color="Green" title="E121">A20</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E133">A32</font></i></a>, <a class="txt" href="glib_001.html#E58"><i><font color="Green" title="E50">Lm21</font></i></a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a></i>
<br/>.= 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E132">A31</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c17">m'aa1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E6:134_1_1_2_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E136">A35</font></i>: 
<font color="Maroon" title="c17">m'aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E7:134_1_1_2_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c17">m'aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a></i>;<br/><a NAME="E8:134_1_1_2_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E137">A36</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">m'aa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E134">A33</font></i></a>, <a class="txt" href="glib_001.html#E4:134_1_1_2_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E135">A34</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green" title="E44">Lm15</font></i></a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a></i>;<br/><a NAME="E9:134_1_1_2_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c17">m'aa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E13:134_1_1_2_1_1_1"><i><font color="Green" title="E124">A23</font></i></a>, <a class="txt" href="glib_001.html#E6:134_1_1_2_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E136">A35</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a></i>;<br/><a NAME="E10:134_1_1_2_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E13:134_1_1_2_1_1_1"><i><font color="Green" title="E124">A23</font></i></a>, <a class="txt" href="glib_001.html#E11:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E131">A30</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E134">A33</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_2_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E137">A36</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green" title="E44">Lm15</font></i></a>, <a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a></i>;<br/><a NAME="E11:134_1_1_2_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_2"><i><font color="Green" title="E111">A9</font></i></a>, <a class="txt" href="glib_001.html#E13:134_1_1_2_1_1_1"><i><font color="Green" title="E124">A23</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E126">A25</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self" title="GLIB_001:def.22">Def22</a></i>;<br/><b>hence </b><a NAME="E12:134_1_1_2_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:134_1_1_2_1_1_1"><i><font color="Green" title="E122">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_1_1_1_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_2_1_1_1_3_1_1_2_1_2"/><i><font color="Green" title="E132">A37</font></i>: 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c17">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:134_1_1_2_1_1_1_3_1_1_2_1_2"/><i><font color="Green" title="E133">A38</font></i>: 
( <font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1"><i><font color="Green" title="E113">A11</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1"><i><font color="Green" title="E114">A12</font></i></a>, <a class="txt" href="glib_001.html#E5:134_1_1_2_1_1_1"><i><font color="Green" title="E116">A14</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E132">A37</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1"><i><font color="Green" title="E113">A11</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1"><i><font color="Green" title="E114">A12</font></i></a>, <a class="txt" href="glib_001.html#E5:134_1_1_2_1_1_1"><i><font color="Green" title="E116">A14</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E132">A37</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><a NAME="E4:134_1_1_2_1_1_1_3_1_1_2_1_2"/><i><font color="Green" title="E134">A39</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_2"><i><font color="Green" title="E111">A9</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E126">A25</font></i></a>, <a class="txt" href="glib_001.html#E11:134_1_1_2_1_1_1_3_1_1"><i><font color="Green" title="E131">A30</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E133">A38</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self" title="GLIB_001:def.22">Def22</a></i>;<br/><a NAME="E5:134_1_1_2_1_1_1_3_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:134_1_1_2_1_1_1"><i><font color="Green" title="E116">A14</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E6:134_1_1_2_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" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E7:134_1_1_2_1_1_1_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_2_1_1_1_3_1_1_2_1_2"><i><font color="Green" title="E134">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:134_1_1_2_1_1_1_3_1_1_2_1_2"/>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:134_1_1_2_1_1_1"><i><font color="Green" title="E122">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:134_1_1_2_1_1_1_3_1_1"/>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_2_1_1_1_3_1_2"/><i><font color="Green" title="E125">A40</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font></span><br/><div class="add"><a NAME="E2:134_1_1_2_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1"><i><font color="Green" title="E112">A10</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_2_1_1_1"><i><font color="Green" title="E119">A18</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E126">A41</font></i>: <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font>
<b>by </b><i><a class="txt" href="glib_001.html#E6:134_1_1_2_1_1_1"><i><font color="Green" title="E117">A15</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2_1_1_1"><i><font color="Green" title="E120">A19</font></i></a>, <a class="txt" href="glib_001.html#E10:134_1_1_2_1_1_1"><i><font color="Green" title="E121">A20</font></i></a>, <a class="txt" href="glib_001.html#E58"><i><font color="Green" title="E50">Lm21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font>
<b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_2"><i><font color="Green" title="E109">A7</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_2_1_1_1"><i><font color="Green" title="E119">A18</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c16">maa1</font> = <font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E5:134_1_1_2_1_1_1_3_1_2"/><i><font color="Green" title="E127">A42</font></i>: 
<font color="Maroon" title="c16">maa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 ;<br/><a NAME="E6:134_1_1_2_1_1_1_3_1_2"/><i><font color="Green" title="E128">A43</font></i>: 
<font color="Maroon" title="c16">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E7:134_1_1_2_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c16">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a></i>;<br/><a NAME="E8:134_1_1_2_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E129">A44</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2_1_1_1"><i><font color="Green" title="E120">A19</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E126">A41</font></i></a>, <a class="txt" href="glib_001.html#E5:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E127">A42</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green" title="E44">Lm15</font></i></a></i>;<br/><a NAME="E9:134_1_1_2_1_1_1_3_1_2"/>
<font color="Maroon" title="c16">maa1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c12">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E13:134_1_1_2"><i><font color="Green" title="E110">A8</font></i></a>, <a class="txt" href="glib_001.html#E6:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E128">A43</font></i></a>, <a class="txt" href="glib_001.html#E59"><i><font color="Green" title="E51">Lm22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:134_1_1_2_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E130">A45</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_2"><i><font color="Green" title="E111">A9</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2_1_1_1"><i><font color="Green" title="E120">A19</font></i></a>, <a class="txt" href="glib_001.html#E5:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E127">A42</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E129">A44</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green" title="E44">Lm15</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_2_1_1_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1_2_1_1_1_3_1_2_2"/><i><font color="Green" title="E131">A46</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:134_1_1_2_1_1_1_3_1_2_2"/><b>then </b><i><font color="Green" title="E132">A47</font></i>: 
<font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_2_1_1_1"><i><font color="Green" title="E115">A13</font></i></a>, <a class="txt" href="glib_001.html#E7:134_1_1_2_1_1_1"><i><font color="Green" title="E118">A16</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E125">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>set </b><font color="Maroon" title="c17">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E3:134_1_1_2_1_1_1_3_1_2_2"/>
( <font color="Maroon" title="c11">Wn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Wn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1"><i><font color="Green" title="E113">A11</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_1"><i><font color="Green" title="E114">A12</font></i></a>, <a class="txt" href="glib_001.html#E5:134_1_1_2_1_1_1"><i><font color="Green" title="E116">A14</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_2_1_1_1_3_1_2_2"><i><font color="Green" title="E132">A47</font></i></a>, <a class="txt" href="glib_001.html#E67"><i><font color="Green" title="E59">Lm30</font></i></a></i>;<br/><a NAME="E4:134_1_1_2_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" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1"><i><font color="Green" title="E112">A10</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_2_1_1_1"><i><font color="Green" title="E123">A22</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E125">A40</font></i></a>, <a class="txt" href="glib_001.html#E10:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E130">A45</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self" title="GLIB_001:def.22">Def22</a></i>;<br/><a NAME="E5:134_1_1_2_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" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:134_1_1_2_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" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E7:134_1_1_2_1_1_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_2_1_1_1"><i><font color="Green" title="E115">A13</font></i></a>, <a class="txt" href="glib_001.html#E7:134_1_1_2_1_1_1"><i><font color="Green" title="E118">A16</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_2"><i><font color="Green" title="E125">A40</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_1_3_1_2_2"><i><font color="Green" title="E131">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E12:134_1_1_2_1_1_1_3_1_2"/>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:134_1_1_2_1_1_1"><i><font color="Green" title="E122">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:134_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_2_1_1_2"/><i><font color="Green" title="E112">A48</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font></span><br/><div class="add"><a NAME="E2:134_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K17" title="GLIB_001:func.17">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> &amp; <font color="Maroon" title="c6">Wn</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Wn</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self" title="GLIB_001:def.20">Def20</a>, <a class="ref" href="glib_001.html#D22" target="_self" title="GLIB_001:def.22">Def22</a></i>;<br/><a NAME="E3:134_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E113">A49</font></i>: 
<font color="Maroon" title="c11">Wn1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">Wn</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E64"><i><font color="Green" title="E56">Lm27</font></i></a></i>;<br/><a NAME="E4:134_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_2"><i><font color="Green" title="E111">A9</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_2_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:134_1_1_2_1_1_2"/>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_2"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_2"><i><font color="Green" title="E108">A6</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2_1_1_2"><i><font color="Green" title="E113">A49</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:134_1_1_2"/>
<font color="Maroon" title="c11">Wn1</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E9:134_1_1"/><b>then </b><i><font color="Green" title="E107">A50</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<a NAME="E10:134_1_1"/><i><font color="Green" title="E108">A51</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_001.html#E7:134_1_1"><i><font color="Green" title="E106">A4</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1"><i><font color="Green" title="E107">A50</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c5">n</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>consider </b><font color="Maroon" title="c6">P</font> being    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E12:134_1_1"/><i><font color="Green" title="E109">A52</font></i>: 
( <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font> &amp; ( for <font color="Olive" title="b1">m</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font> holds <br/><font color="Maroon" title="c6">P</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E10:134_1_1"><i><font color="Green" title="E108">A51</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> &amp; <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> )</span><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" title="c7">m</font> be  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font> implies <font color="Maroon" title="c6">P</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font> )</span><br/><b>assume </b><a NAME="E1:134_1_1_3"/><i><font color="Green" title="E110">A53</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">P</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font></span><br/><a NAME="E2:134_1_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1"><i><font color="Green" title="E109">A52</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E3:134_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E4:134_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E5:134_1_1_3"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3"><i><font color="Green" title="E110">A53</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:134_1_1_3"/>
<font color="Maroon" title="c6">P</font> <a href="glib_001.html#K19" title="GLIB_001:func.19">.rfind</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1"><i><font color="Green" title="E109">A52</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_3"><i><font color="Green" title="E110">A53</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E14:134_1_1"/><b>then </b>
<font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 
<b>by </b><i><a class="txt" href="glib_001.html#E104"><i><font color="Green" title="E96">Lm67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:134_1_1"/>
( <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> &amp; <font color="Maroon" title="c6">P</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
