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

<b>let </b><font color="Maroon" title="c1">f</font> be   <a href="sppol_2.html#NM1" title="SPPOL_2:NM.1">S-Sequence_in_R2</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="E1:52"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:52"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E4:52"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E5:52"/><i><font color="Green" title="E45">A4</font></i>: 
( 0 <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="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> )
 
<b>by </b><i><a class="txt" href="jordan1g.html#E3:52"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:52"/><b>then </b><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:52"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E8:52"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T18" title="SPPOL_2:th.18">SPPOL_2:18</a></i>;<br/>
<a NAME="E9:52"/><b>then </b><i><font color="Green" title="E47">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font> &amp;  <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</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="c1">f</font> )
 
<b>by </b><i><a class="txt" href="jordan1g.html#E1:52"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="jordan3.html#T41" title="JORDAN3:th.41">JORDAN3:41</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> )
 </span><b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</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="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/><i><font color="Green" title="E48">A7</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1"/><b>then </b><i><font color="Green" title="E49">A8</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E4:52"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="jordan3.html#T45" title="JORDAN3:th.45">JORDAN3:45</a></i>;<br/><a NAME="E3:52_1_1"/>
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="c1">f</font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:52_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:52_1_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E3:52"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="jordan1g.html#E4:52"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="jordan1g.html#E1:52_1_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E6:52_1_1"/><b>then </b><i><font color="Green" title="E50">A9</font></i>: 
 <a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</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="c1">f</font>,1,<span class="p2">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan3.html#D5" title="JORDAN3:def.5">JORDAN3:def 5</a></i>;<br/><i><font color="Green" title="E51">A10</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="c1">f</font>,1,<font color="Maroon" title="c3">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</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="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="jordan4.html#T20" title="JORDAN4:th.20">JORDAN4:20</a></i>
<br/>.= 
<font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/><i><font color="Green" title="E52">A11</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="c1">f</font>,1,<span class="p2">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">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="txt" href="jordan1g.html#E9:52"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="jordan4.html#T20" title="JORDAN4:th.20">JORDAN4:20</a></i>
<br/>.= 
<font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan1g.html#E9:52"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="binarith.html#T56" title="BINARITH:th.56">BINARITH:56</a></i>
;<br/><b>then </b><i><font color="Green" title="E53">A12</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c1">f</font>,1,<span class="p3">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</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="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>
<br/>.= 
<font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:52_1_1_4"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">i</font>
 ;<br/><a NAME="E2:52_1_1_4"/><b>then </b><i><font color="Green" title="E54">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_4_1_1"/>
( <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> )
 </span><b>by </b><i><a class="txt" href="jordan1g.html#E2:52_1_1_4"><i><font color="Green" title="E54">A13</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="52_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_4_1_1_1"/>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E55">A14</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E2:52_1_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:52_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E6:52"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E4:52_1_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E56">A15</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<span class="p2">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1g.html#E8:52_1_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1_4"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> = 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1_4"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="jordan3.html#T32" title="JORDAN3:th.32">JORDAN3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<span class="p2">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E9:52"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1_4"><i><font color="Green" title="E54">A13</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1_4_1_1_1"><i><font color="Green" title="E55">A14</font></i></a>, <a class="ref" href="jordan3.html#T32" title="JORDAN3:th.32">JORDAN3:32</a></i>
<br/>.= 
<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="c1">f</font>,1,<span class="p3">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E4:52_1_1_4_1_1_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_4_1_1_2"/><i><font color="Green" title="E55">A16</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_4_1_1_2"/><i><font color="Green" title="E56">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> = 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1_4"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="jordan3.html#T32" title="JORDAN3:th.32">JORDAN3:32</a></i>
<br/>.= 
<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="c1">f</font>,1,<span class="p3">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="jordan1g.html#E4:52"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="jordan1g.html#E8:52_1_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="txt" href="jordan1g.html#E1:52_1_1_4_1_1_2"><i><font color="Green" title="E55">A16</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_1_4_1_1_2"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:52_1_1_4"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<font color="Maroon" title="c3">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">f</font>,1,<span class="p3">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 ;<br/></div><b>end;</b></div><a NAME="E11:52_1_1"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="jordan1g.html#E6:52_1_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="jordan1g.html#E7:52_1_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="jordan1g.html#E9:52_1_1"><i><font color="Green" title="E53">A12</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/><b>hence </b><a NAME="E12:52_1_1"/>
 <a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1g.html#E3:52"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="jordan1g.html#E4:52"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="finseq_5.html#T12" title="FINSEQ_5:th.12">FINSEQ_5:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green" title="E48">A18</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><a NAME="E2:52_1_2"/><b>then </b><i><font color="Green" title="E49">A19</font></i>: 
 <a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan1g.html#E4:52"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="jordan3.html#D5" title="JORDAN3:def.5">JORDAN3:def 5</a></i>;<br/><a NAME="E3:52_1_2"/><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 <b>by </b><i><a class="txt" href="jordan1g.html#E3:52"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="jordan1g.html#E4:52"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="jordan1g.html#E1:52_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:52_1_2"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#T47" title="FINSEQ_6:th.47">FINSEQ_6:47</a></i>;<br/><b>hence </b><a NAME="E5:52_1_2"/>
 <a href="jordan3.html#K4" title="JORDAN3:func.4">R_Cut</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c1">f</font>,1,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K5" title="FINSEQ_4:func.5">..</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1g.html#E5:52"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="jordan1g.html#E1:52_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="jordan1g.html#E2:52_1_2"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="jordan1g.html#E3:52_1_2"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="jordan4.html#T27" title="JORDAN4:th.27">JORDAN4:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
