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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>2</sub></font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) ) or ( <font color="Maroon">a<sub>2</sub></font> is  not    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> ) );<br/>
<div><i><font color="Green">E108</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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>4</sub></font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> is    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E3:134_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>]
 ;<br/><b>hence </b><a NAME="E4:134_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is  not    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:134_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E2:134_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:134_1_1"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>] holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font>
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:134_1_1"/><i><font color="Green">E110</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S1">RECDEF_1:sch 1</a>(<a class="txt" href="glib_001.html#E1:134_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1"><i><font color="Green">E109</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:134_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="glib_001.html#E113"><i><font color="Green">Lemma105</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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:134_1_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="glib_001.html#M3" target="_self">Walk</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:134_1_1_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1"><i><font color="Green">E110</font></i></a></i>;<br/><b>thus </b><a NAME="E4:134_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="glib_001.html#E114"><i><font color="Green">Lemma106</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:134_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E8:134_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="glib_001.html#E5:134_1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="glib_001.html#E7:134_1_1"><i><font color="Green">E112</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 as    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) );<br/>
<a NAME="E10:134_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/>
<a NAME="E11:134_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1"><i><font color="Green">E110</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:134_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>5</sub></font>]
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:134_1_1_3"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K20" target="_self">.remove</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="glib_001.html#E114"><i><font color="Green">Lemma106</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E5:134_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1"><i><font color="Green">E110</font></i></a></i>;<br/><b>hence </b><a NAME="E6:134_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E114</font></i></a></i>;<br/><a NAME="E7:134_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E67"><i><font color="Green">Lemma59</font></i></a></i>;<br/><b>hence </b><a NAME="E8:134_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font>;<br/><b>assume </b><a NAME="E9:134_1_1_3"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/><a NAME="E10:134_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:134_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E12:134_1_1_3"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:134_1_1_3"/><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E67"><i><font color="Green">Lemma59</font></i></a></i>;<br/><a NAME="E14:134_1_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_3_1_1"/>
( <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E120</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a></i>;<br/><a NAME="E3:134_1_1_3_1_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a></i>;<br/><a NAME="E4:134_1_1_3_1_1_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="glib_001.html#E91"><i><font color="Green">Lemma83</font></i></a>, <a class="txt" href="glib_001.html#E92"><i><font color="Green">Lemma84</font></i></a></i>;<br/><a NAME="E5:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K7" target="_self">.append</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="glib_001.html#D12" target="_self">Def12</a></i>;<br/><div><i><font color="Green">E125</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1_3_1_1_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:134_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:134_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E114</font></i></a></i>;<br/><b>hence </b><a NAME="E4:134_1_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a></i>;<br/></div><b>end;</b></div><a NAME="E8:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_3_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:134_1_1_3_1_1_1"/><i><font color="Green">E127</font></i>: 
( not 1 is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><i><font color="Green">E128</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  = 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a> 
<b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green">Lemma49</font></i></a></i>
;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font>;<br/><a NAME="E11:134_1_1_3_1_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_001.html#E92"><i><font color="Green">Lemma84</font></i></a></i>;<br/><a NAME="E12:134_1_1_3_1_1_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a></i>;<br/><a NAME="E13:134_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E13:134_1_1_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_3_1_1_1_3_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_3"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a></i>;<br/><a NAME="E3:134_1_1_3_1_1_1_3_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_3_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><b>then </b><i><font color="Green">E134</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="glib_001.html#E6:134_1_1_3_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_001.html#E10:134_1_1_3_1_1_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="glib_001.html#E62"><i><font color="Green">Lemma54</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_3"><i><font color="Green">E116</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_3_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="glib_001.html#E61"><i><font color="Green">Lemma53</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E7:134_1_1_3_1_1_1_3_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E8:134_1_1_3_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E9:134_1_1_3_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E134</font></i></a>, <a class="txt" href="glib_001.html#E6:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E135</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma48</font></i></a></i>;<br/><a NAME="E10:134_1_1_3_1_1_1_3_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_001.html#E7:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E136</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E11:134_1_1_3_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_001.html#E6:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E135</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E137</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma48</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:134_1_1_3_1_1_1_3_1_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E3:134_1_1_3_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E141</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then </b><i><font color="Green">E142</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E6:134_1_1_3_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="txt" href="glib_001.html#E10:134_1_1_3_1_1_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E140</font></i></a>, <a class="txt" href="glib_001.html#E62"><i><font color="Green">Lemma54</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E139</font></i></a>, <a class="txt" href="glib_001.html#E61"><i><font color="Green">Lemma53</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E141</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E8:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E142</font></i></a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E141</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma48</font></i></a></i>;<br/><a NAME="E9:134_1_1_3_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E143</font></i></a>, <a class="txt" href="glib_001.html#E13:134_1_1_3_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E10:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="txt" href="glib_001.html#E13:134_1_1_3_1_1_1"><i><font color="Green">E131</font></i></a>, <a class="txt" href="glib_001.html#E11:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E138</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E141</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_3_1_1_1_3_1_1_2_1_1"><i><font color="Green">E144</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma48</font></i></a></i>;<br/><a NAME="E11:134_1_1_3_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="txt" href="glib_001.html#E13:134_1_1_3_1_1_1"><i><font color="Green">E131</font></i></a></i>;<br/><b>hence </b><a NAME="E12:134_1_1_3_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:134_1_1_3_1_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_1_2_1_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:134_1_1_3_1_1_1_3_1_1_2_1_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is   <a href="nat_1.html#NM1">Nat</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E5:134_1_1_3_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_1_2_1_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E5:134_1_1_3_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_1_2_1_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></a></i>;<br/><a NAME="E4:134_1_1_3_1_1_1_3_1_1_2_1_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1_3_1_1_2_1_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="glib_001.html#E11:134_1_1_3_1_1_1_3_1_1"><i><font color="Green">E138</font></i></a></i>;<br/><a NAME="E5:134_1_1_3_1_1_1_3_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:134_1_1_3_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:134_1_1_3_1_1_1_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:134_1_1_3_1_1_1_3_1_1_2_1_2"><i><font color="Green">E141</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:134_1_1_3_1_1_1_3_1_1_2_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:134_1_1_3_1_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:134_1_1_3_1_1_1_3_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_1_3_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_3_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><b>then </b><i><font color="Green">E133</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="glib_001.html#E6:134_1_1_3_1_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_001.html#E10:134_1_1_3_1_1_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="glib_001.html#E62"><i><font color="Green">Lemma54</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="glib_001.html#E12:134_1_1_3"><i><font color="Green">E116</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_3_1_1_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="glib_001.html#E61"><i><font color="Green">Lemma53</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E5:134_1_1_3_1_1_1_3_1_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:134_1_1_3_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E8:134_1_1_3_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="glib_001.html#E5:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma48</font></i></a></i>;<br/><a NAME="E9:134_1_1_3_1_1_1_3_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_001.html#E6:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E10:134_1_1_3_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3_1_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="glib_001.html#E5:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="glib_001.html#E8:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E136</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma48</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_3_1_1_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:134_1_1_3_1_1_1_3_1_2_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E7:134_1_1_3_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="glib_001.html#E4:134_1_1_3_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E132</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E3:134_1_1_3_1_1_1_3_1_2_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is   <a href="nat_1.html#NM1">Nat</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E5:134_1_1_3_1_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="glib_001.html#E2:134_1_1_3_1_1_1_3_1_2_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></a></i>;<br/><a NAME="E4:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="glib_001.html#E10:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E137</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="txt" href="glib_001.html#E12:134_1_1_3_1_1_1"><i><font color="Green">E130</font></i></a></i>;<br/><a NAME="E5:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E6:134_1_1_3_1_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E7:134_1_1_3_1_1_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_2_2"><i><font color="Green">E138</font></i></a>, <a class="txt" href="glib_001.html#E7:134_1_1_3_1_1_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="glib_001.html#E4:134_1_1_3_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_1_3_1_2"><i><font color="Green">E132</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:134_1_1_3_1_1_1_3_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:134_1_1_3_1_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:134_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="134_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:134_1_1_3_1_1_2"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:134_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K17" target="_self">.find</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</a></i>;<br/><a NAME="E3:134_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E68"><i><font color="Green">Lemma60</font></i></a></i>;<br/><a NAME="E4:134_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_3_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:134_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:134_1_1_3_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="glib_001.html#E3:134_1_1_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="glib_001.html#E9:134_1_1_3"><i><font color="Green">E115</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:134_1_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E13:134_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E14:134_1_1"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="glib_001.html#E11:134_1_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="glib_001.html#E13:134_1_1"><i><font color="Green">E114</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="glib_001.html#M4" target="_self">Subwalk</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E16:134_1_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E14:134_1_1"><i><font color="Green">E115</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:134_1_1_4"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:134_1_1_4"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E16:134_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:134_1_1_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:134_1_1_4"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:134_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:134_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E1:134_1_1_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:134_1_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="glib_001.html#K19" target="_self">.rfind</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E16:134_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="glib_001.html#E1:134_1_1_4"><i><font color="Green">E117</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:134_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 
<b>by </b><i><a class="txt" href="glib_001.html#E109"><i><font color="Green">Lemma101</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:134_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="glib_001.html#V5" target="_self">Path-like</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self">Def28</a></i>;<br/>


</div>
