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

<a NAME="E1:39_1_2"/><i><font color="Green">E92</font></i>: 
1 <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:39_1_2"/>
for <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( <font color="Olive">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k1</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k1</font> ) ) ) &amp; ( <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) &amp; ( <font color="Olive">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k2</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">k2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k2</font> ) ) ) &amp; ( <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) holds <br/><font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">k2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_2_1"/><i><font color="Green">E93</font></i>: 
( ( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font> ) ) ) &amp; ( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) &amp; ( <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font> ) ) ) &amp; ( <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">g</font> <a href="finseq_8.html#R2" target="_self">is_substring_of</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_2_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
0 <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">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:39_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_2_1_1_1_1_1_1"/>
( ( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_1_1_1_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:39_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_8.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:39_1_2_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i/>;<br/><a NAME="E5:39_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i>, <a class="ref" href="finseq_8.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E6:39_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T7" target="_self">Th7</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="39_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_1_1_1_2"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E3:39_1_2_1_1_1_1_1_1_2"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> holds <br/> not <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_2_1_1_1_1_1_1_2_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:39_1_2_1_1_1_1_1_1_2_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:39_1_2_1_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E4:39_1_2_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E5:39_1_2_1_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>then </b><i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self">Th4</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<a NAME="E7:39_1_2_1_1_1_1_1_1_2_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, , <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
;<br/>
<a NAME="E9:39_1_2_1_1_1_1_1_1_2_1"/><b>then </b>
(  not 1 <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">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> or  not  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_8.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_8.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_2_1_1_1_1_1_1_2_1"/>
not <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i><a class="ref" href="finseq_8.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:39_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_8.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E5:39_1_2_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_8.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_1_1_1_3"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E3:39_1_2_1_1_1_1_1_1_3"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> holds <br/> not <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_2_1_1_1_1_1_1_3_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:39_1_2_1_1_1_1_1_1_3_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:39_1_2_1_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E4:39_1_2_1_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E5:39_1_2_1_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>then </b><i><font color="Green">E111</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self">Th4</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<a NAME="E7:39_1_2_1_1_1_1_1_1_3_1"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_8.html#D2" target="_self">Def2</a>, , <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
;<br/>
<a NAME="E9:39_1_2_1_1_1_1_1_1_3_1"/><b>then </b>
(  not 1 <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">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> or  not  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_8.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_8.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_8.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_2_1_1_1_1_1_1_3_1"/>
not <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i><a class="ref" href="finseq_8.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:39_1_2_1_1_1_1_1_1_3"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_8.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E5:39_1_2_1_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_8.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_1_1_1_4"/>
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_2_1_1_1_1_1_1_4"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:39_1_2_1_1_1_1"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_2"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_2_1_1_1_2_1_1"/>
( ( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_2_1_1_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:39_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:39_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_8.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:39_1_2_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i/>;<br/><a NAME="E5:39_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 <b>by </b><i>, <a class="ref" href="finseq_8.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E6:39_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#T6" target="_self">Th6</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="39_1_2_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_2_1_1_2"/>
( <font color="Maroon">k1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_2_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_2_1_1_3"/>
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_2_1_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2_1_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:39_1_2_1_1_1_2_1_1_4"/>
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_1_2_1_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:39_1_2_1_1_1_2"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_1_2_1"/>
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font>
 ;<br/>


</div><b>end;</b></div>


</div>
