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

<b>let </b><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">i2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>









<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E55">A1</font></i>: 
( <font color="Maroon" title="c1">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> )
 ;<br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E56">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> &amp; <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c2">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> )
 
<b>by </b><i><a class="ref" href="jordan4.html#D2" target="_self" title="JORDAN4:def.2">Def2</a></i>;<br/>
<a NAME="E3:59"/><i><font color="Green" title="E57">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:59"/><b>then </b><i><font color="Green" title="E58">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:59"/><b>then </b><i><font color="Green" title="E59">A5</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E6:59"/><i><font color="Green" title="E60">A6</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:59"/><i><font color="Green" title="E61">A7</font></i>: 
<font color="Maroon" title="c2">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:59"/><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1"/>
( <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> or <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_1"/><i><font color="Green" title="E63">A9</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_1_1_1_1_1"/>
( <font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i2</font> or <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i2</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_1"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2"/><i><font color="Green" title="E64">A10</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E65">A11</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c2">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="jordan4.html#T40" target="_self" title="JORDAN4:th.40">Th40</a></i>;<br/><a NAME="E3:59_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E66">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2"/><i><font color="Green" title="E67">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="jordan4.html#T21" target="_self" title="JORDAN4:th.21">Th21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_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="E1:59_1_1_1_1_1_2_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E68">A14</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E69">A15</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:59_1_1_1_1_1_2_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E68">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E70">A16</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><i><font color="Green" title="E71">A17</font></i>: <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c2">i2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="jordan4.html#T40" target="_self" title="JORDAN4:th.40">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E70">A16</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E4:59"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E68">A14</font></i></a>, <a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E70">A16</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E69">A15</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/><a NAME="E6:59_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:59_1_1_1_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:59_1_1_1_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E9:59_1_1_1_1_1_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E72">A18</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><i><font color="Green" title="E73">A19</font></i>: <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="jordan4.html#T37" target="_self" title="JORDAN4:th.37">Th37</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="jordan4.html#E10:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E72">A18</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/><a NAME="E12:59_1_1_1_1_1_2_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E68">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E74">A20</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:59_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E75">A21</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><div><i><font color="Green" title="E76">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_1_1_1_1_2_1_1_1_3"/><i><font color="Green" title="E77">A23</font></i>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 ;<br/><a NAME="E2:59_1_1_1_1_1_2_1_1_1_3"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1_1_2_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_1_3"><i><font color="Green" title="E77">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_1_1_1_2_1_1_1_4_1"/>
( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 or <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E15:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E76">A22</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_4_1_1"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_1_4_1_1"/><i><font color="Green" title="E77">A24</font></i>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E78">A25</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E14:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E75">A21</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_1_4_1_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="jordan4.html#D2" target="_self" title="JORDAN4:def.2">Def2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E79">A26</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T39" title="GOBOARD7:th.39">GOBOARD7:39</a></i>;<br/><a NAME="E6:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2_1_1_1_4_1_1"><i><font color="Green" title="E78">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_1_1_1_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E80">A27</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E14:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E75">A21</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_1_1_1_2_1_1_1_4_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E71">A17</font></i></a>, <a class="txt" href="jordan4.html#E11:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E73">A19</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_1_4_1_1"><i><font color="Green" title="E79">A26</font></i></a>, <a class="txt" href="jordan4.html#E7:59_1_1_1_1_1_2_1_1_1_4_1_1"><i><font color="Green" title="E80">A27</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_1_4_1_2"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_1_4_1_2"/><i><font color="Green" title="E77">A28</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green" title="E78">A29</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E13:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E74">A20</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_1_4_1_2"><i><font color="Green" title="E77">A28</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green" title="E79">A30</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T38" title="GOBOARD7:th.38">GOBOARD7:38</a></i>;<br/><a NAME="E6:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2_1_1_1_4_1_2"><i><font color="Green" title="E78">A29</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_1_1_1_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green" title="E80">A31</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2_1_1_1_4_1_2"><i><font color="Green" title="E78">A29</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_1_1_1_2_1_1_1_4_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E71">A17</font></i></a>, <a class="txt" href="jordan4.html#E11:59_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E73">A19</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_1_4_1_2"><i><font color="Green" title="E79">A30</font></i></a>, <a class="txt" href="jordan4.html#E7:59_1_1_1_1_1_2_1_1_1_4_1_2"><i><font color="Green" title="E80">A31</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:59_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E68">A32</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font>
 ;<br/><div class="add"><i><font color="Green" title="E69">A33</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="jordan4.html#T21" target="_self" title="JORDAN4:th.21">Th21</a></i>
<br/>.= 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E68">A32</font></i></a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><i><font color="Green" title="E70">A34</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">i2</font>
<b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="jordan4.html#T40" target="_self" title="JORDAN4:th.40">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font>
<b>by </b><i><a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E68">A32</font></i></a>, <a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
;<br/><a NAME="E4:59_1_1_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:59_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E71">A35</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">i2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><i><font color="Green" title="E72">A36</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="jordan4.html#T21" target="_self" title="JORDAN4:th.21">Th21</a></i>
<br/>.= 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E68">A32</font></i></a>, <a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><i><font color="Green" title="E73">A37</font></i>: <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c2">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2"><i><font color="Green" title="E65">A11</font></i></a>, <a class="txt" href="jordan4.html#E2:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E69">A33</font></i></a>, <a class="txt" href="jordan4.html#E3:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E70">A34</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E71">A35</font></i></a>, <a class="ref" href="jordan3.html#T15" title="JORDAN3:th.15">JORDAN3:15</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E6:59"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="jordan4.html#E6:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E72">A36</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><a NAME="E8:59_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E10:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E11:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:59_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E74">A38</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><i><font color="Green" title="E75">A39</font></i>: <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="jordan4.html#T37" target="_self" title="JORDAN4:th.37">Th37</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="jordan4.html#E12:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E74">A38</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/><a NAME="E14:59_1_1_1_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4
 <b>by </b><i><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a></i>;<br/><a NAME="E15:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:59_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E68">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:59_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E76">A40</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E6:59"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E68">A32</font></i></a>, <a class="ref" href="goboard7.html#T39" title="GOBOARD7:th.39">GOBOARD7:39</a></i>;<br/><a NAME="E18:59_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E77">A41</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E6:59"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="jordan4.html#E4:59_1_1_1_1_1_2"><i><font color="Green" title="E66">A12</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E19:59_1_1_1_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E20:59_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E7:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E73">A37</font></i></a>, <a class="txt" href="jordan4.html#E13:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E75">A39</font></i></a>, <a class="txt" href="jordan4.html#E17:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E76">A40</font></i></a>, <a class="txt" href="jordan4.html#E18:59_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E77">A41</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:59_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_2"/><i><font color="Green" title="E63">A42</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_2"/><b>then </b><i><font color="Green" title="E64">A43</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<font color="Maroon" title="c2">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="jordan4.html#T38" target="_self" title="JORDAN4:th.38">Th38</a></i>;<br/><a NAME="E3:59_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:59_1_1_2"/><b>then </b><i><font color="Green" title="E65">A44</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:59_1_1_2"/><i><font color="Green" title="E66">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:59_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_2"/><b>then </b><i><font color="Green" title="E67">A46</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E6:59"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_2"><i><font color="Green" title="E66">A45</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E8:59_1_1_2"/><i><font color="Green" title="E68">A47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><div><i><font color="Green" title="E69">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_1_1_2_1_1"/>
( <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E5:59_1_1_2"><i><font color="Green" title="E66">A45</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:59_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><div class="add"><a NAME="E2:59_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:59_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E70">A49</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:59_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E71">A50</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:59_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E3:59_1_1_2_1_1_1"><i><font color="Green" title="E70">A49</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:59_1_1_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i1</font></span>)</span>
 ;<br/><a NAME="E7:59_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E72">A51</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E7:59_1_1_2"><i><font color="Green" title="E67">A46</font></i></a>, <a class="txt" href="jordan4.html#E8:59_1_1_2"><i><font color="Green" title="E68">A47</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<font color="Maroon" title="c2">i2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="ref" href="jordan4.html#T38" target="_self" title="JORDAN4:th.38">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E7:59_1_1_2_1_1_1"><i><font color="Green" title="E72">A51</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E6:59"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="jordan4.html#E8:59_1_1_2"><i><font color="Green" title="E68">A47</font></i></a>, <a class="txt" href="jordan4.html#E4:59_1_1_2_1_1_1"><i><font color="Green" title="E71">A50</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:59_1_1_2_1_1_2"/><i><font color="Green" title="E70">A52</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E71">A53</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E6:59"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><i><font color="Green" title="E72">A54</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">i2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2_1_1_2"><i><font color="Green" title="E70">A52</font></i></a>, <a class="ref" href="jordan4.html#T38" target="_self" title="JORDAN4:th.38">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">i2</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>

<br/>.= 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">i2</font>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/><a NAME="E4:59_1_1_2_1_1_2"/><i><font color="Green" title="E73">A55</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E74">A56</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E5:59"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="jordan4.html#E6:59"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2_1_1_2"><i><font color="Green" title="E70">A52</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
1

;<br/><b>thus </b><font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,1,<font color="Maroon" title="c2">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59_1_1_2"><i><font color="Green" title="E64">A43</font></i></a>, <a class="txt" href="jordan4.html#E2:59_1_1_2_1_1_2"><i><font color="Green" title="E71">A53</font></i></a>, <a class="txt" href="jordan4.html#E3:59_1_1_2_1_1_2"><i><font color="Green" title="E72">A54</font></i></a>, <a class="txt" href="jordan4.html#E4:59_1_1_2_1_1_2"><i><font color="Green" title="E73">A55</font></i></a>, <a class="ref" href="jordan3.html#T15" title="JORDAN3:th.15">JORDAN3:15</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E4:59"><i><font color="Green" title="E58">A4</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_2_1_1_2"><i><font color="Green" title="E74">A56</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E3:59"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2_1_1_2"><i><font color="Green" title="E70">A52</font></i></a>, <a class="ref" href="jordan4.html#T14" target="_self" title="JORDAN4:th.14">Th14</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:59_1_1_2"/>
<font color="Maroon" title="c2">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:59_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">i2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:59_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E13:59_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">i2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:59_1_1_2"/><b>then </b><i><font color="Green" title="E70">A57</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><i><font color="Green" title="E71">A58</font></i>: <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="ref" href="jordan4.html#T39" target="_self" title="JORDAN4:th.39">Th39</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E7:59"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="jordan4.html#E8:59"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="txt" href="jordan4.html#E14:59_1_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_2"><i><font color="Green" title="E65">A44</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/><a NAME="E16:59_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:59_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:59_1_1_2"/><b>then </b><i><font color="Green" title="E72">A59</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E19:59_1_1_2"/><b>then </b><i><font color="Green" title="E73">A60</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><div><i><font color="Green" title="E74">A61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_1_2_3"/><i><font color="Green" title="E75">A62</font></i>: 
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 ;<br/><a NAME="E2:59_1_1_2_3"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a>, <a class="txt" href="jordan4.html#E1:59_1_1_2_3"><i><font color="Green" title="E75">A62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_2_4_1"/>
( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 or <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E20:59_1_1_2"><i><font color="Green" title="E74">A61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_4_1_1"><b>case </b></a><a NAME="E1:59_1_1_2_4_1_1"/><i><font color="Green" title="E75">A63</font></i>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/><div class="add"><a NAME="E2:59_1_1_2_4_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:59_1_1_2_4_1_1"/><b>then </b>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_2_4_1_1"/><b>then </b><i><font color="Green" title="E76">A64</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E1:59"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="jordan4.html#E19:59_1_1_2"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2_4_1_1"><i><font color="Green" title="E75">A63</font></i></a>, <a class="ref" href="jordan4.html#D2" target="_self" title="JORDAN4:def.2">Def2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_2_4_1_1"/><b>then </b><i><font color="Green" title="E77">A65</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T39" title="GOBOARD7:th.39">GOBOARD7:39</a></i>;<br/><a NAME="E6:59_1_1_2_4_1_1"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_2_4_1_1"><i><font color="Green" title="E76">A64</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_2_4_1_1"/><b>then </b><i><font color="Green" title="E78">A66</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E19:59_1_1_2"><i><font color="Green" title="E73">A60</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_2_4_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_2_4_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E9:59_1_1_2"><i><font color="Green" title="E69">A48</font></i></a>, <a class="txt" href="jordan4.html#E15:59_1_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_2_4_1_1"><i><font color="Green" title="E77">A65</font></i></a>, <a class="txt" href="jordan4.html#E7:59_1_1_2_4_1_1"><i><font color="Green" title="E78">A66</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2_4_1_2"><b>case </b></a><a NAME="E1:59_1_1_2_4_1_2"/><i><font color="Green" title="E75">A67</font></i>: 
<font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ;<br/><div class="add"><a NAME="E2:59_1_1_2_4_1_2"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:59_1_1_2_4_1_2"/><b>then </b>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:59_1_1_2_4_1_2"/><b>then </b><i><font color="Green" title="E76">A68</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E18:59_1_1_2"><i><font color="Green" title="E72">A59</font></i></a>, <a class="txt" href="jordan4.html#E1:59_1_1_2_4_1_2"><i><font color="Green" title="E75">A67</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E5:59_1_1_2_4_1_2"/><b>then </b><i><font color="Green" title="E77">A69</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T38" title="GOBOARD7:th.38">GOBOARD7:38</a></i>;<br/><a NAME="E6:59_1_1_2_4_1_2"/>
<font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_2_4_1_2"><i><font color="Green" title="E76">A68</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:59_1_1_2_4_1_2"/><b>then </b><i><font color="Green" title="E78">A70</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:59_1_1_2_4_1_2"><i><font color="Green" title="E76">A68</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E8:59_1_1_2_4_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_2_4_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan4.html#E2:59"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="jordan4.html#E9:59_1_1_2"><i><font color="Green" title="E69">A48</font></i></a>, <a class="txt" href="jordan4.html#E15:59_1_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="jordan4.html#E5:59_1_1_2_4_1_2"><i><font color="Green" title="E77">A69</font></i></a>, <a class="txt" href="jordan4.html#E7:59_1_1_2_4_1_2"><i><font color="Green" title="E78">A70</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:59_1_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:59"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ;<br/>


</div>
