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

<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</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>let </b><font color="Maroon" title="c3">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><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="topreal1.html#NR2" title="TOPREAL1:NR.2">is_S-Seq</a> 
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c3">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>and </b><br/><a NAME="E4:37"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c2">g</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="c3">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="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green" title="E35">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">i</font> )
 <b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></i>;<br/>
<a NAME="E7:37"/><i><font color="Green" title="E36">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="c3">p</font>,<font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan3.html#T41" target="_self" title="JORDAN3:th.41">Th41</a></i>;<br/>
<a NAME="E8:37"/><i><font color="Green" title="E37">A7</font></i>: 
 <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</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="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan3.html#T41" target="_self" title="JORDAN3:th.41">Th41</a></i>;<br/>
<a NAME="E9:37"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E10:37"/><b>then </b><i><font color="Green" title="E38">A8</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="c1">f</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E6:37"><i><font color="Green" title="E35">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="E11:37"/><b>then </b><i><font color="Green" title="E39">A9</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="c3">p</font>,<font color="Maroon" title="c1">f</font></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"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">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#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="jordan3.html#T27" target="_self" title="JORDAN3:th.27">Th27</a></i>;<br/>
<a NAME="E12:37"/><b>then </b><i><font color="Green" title="E40">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,<span class="p2">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E13:37"/>
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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E14:37"/><b>then </b><i><font color="Green" title="E41">A11</font></i>: 
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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<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="E15:37"/>
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="c3">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> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="jordan3.html#T17" target="_self" title="JORDAN3:th.17">Th17</a></i>;<br/>
<a NAME="E16:37"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E10:37"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="jordan3.html#T27" target="_self" title="JORDAN3:th.27">Th27</a></i>;<br/>
<a NAME="E17:37"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E10:37"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<i><font color="Green" title="E43">A13</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E19:37"/><b>then </b><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E20:37"/><i><font color="Green" title="E45">A15</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E21:37"/><i><font color="Green" title="E46">A16</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E22:37"/><i><font color="Green" title="E47">A17</font></i>: 
for <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">n2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n1</font> &amp; <font color="Olive" title="b1">n1</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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n2</font> &amp; <font color="Olive" title="b2">n2</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> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n2</font> holds <br/><font color="Olive" title="b1">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n1</font>, <font color="Maroon" title="c6">n2</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>assume </b><a NAME="E1:37_2"/><i><font color="Green" title="E48">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n1</font> &amp; <font color="Maroon" title="c5">n1</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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n2</font> &amp; <font color="Maroon" title="c6">n2</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> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n2</font> )
 ;<br/>

<a NAME="E2:37_2"/><b>then </b>
( <font color="Maroon" title="c5">n1</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> &amp; <font color="Maroon" title="c6">n2</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>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E3:37_2"/>
<font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E21:37"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="jordan3.html#E1:37_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:37"/><i><font color="Green" title="E48">A19</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</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="c2">g</font> &amp; <font color="Olive" title="b2">x2</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="c2">g</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_3"/><i><font color="Green" title="E49">A20</font></i>: 
( <font color="Maroon" title="c5">x1</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="c2">g</font> &amp; <font color="Maroon" title="c6">x2</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="c2">g</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">n1</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">n2</font> = <font color="Maroon" title="c6">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:37_3"/><i><font color="Green" title="E50">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n1</font> &amp; <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n2</font> &amp; <font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> )
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1"><b>now </b></a><div class="add"><i><font color="Green" title="E51">A22</font></i>: <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c2">g</font></span>)</span> = 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E14:37"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_3_1_2_1"/>
( <font color="Maroon" title="c7">n1</font> <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="c2">g</font> or <font color="Maroon" title="c8">n2</font> <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="c2">g</font> or ( <font color="Maroon" title="c7">n1</font> <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="c2">g</font> &amp; <font color="Maroon" title="c8">n2</font> <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="c2">g</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_2_1_1"><b>case </b></a><a NAME="E1:37_3_1_2_1_1"/><i><font color="Green" title="E52">A23</font></i>: 
<font color="Maroon" title="c7">n1</font> <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="c2">g</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_3_1_2_1_1_1"/><i><font color="Green" title="E53">A24</font></i>: 
<font color="Maroon" title="c8">n2</font> <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="c2">g</font>
 ;<br/><a NAME="E2:37_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:37_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E54">A25</font></i>: 
<font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><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="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:37_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n2</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="c3">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="c8">n2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:37_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">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="c8">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E10:37"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3_1_2_1_1_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="jordan3.html#T27" target="_self" title="JORDAN3:th.27">Th27</a></i>;<br/><a NAME="E6:37_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E55">A26</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3_1"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3_1_2_1_1"><i><font color="Green" title="E52">A23</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E7:37_3_1_2_1_1_1"/><i><font color="Green" title="E56">A27</font></i>: 
<font color="Maroon" title="c8">n2</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="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3_1_2_1_1_1"><i><font color="Green" title="E54">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="E8:37_3_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">n2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E3:37"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E6:37_3_1_2_1_1_1"><i><font color="Green" title="E55">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:37_3_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jordan3.html#E6:37_3_1_2_1_1_1"><i><font color="Green" title="E55">A26</font></i></a>, <a class="txt" href="jordan3.html#E7:37_3_1_2_1_1_1"><i><font color="Green" title="E56">A27</font></i></a>, <a class="ref" href="jordan3.html#T45" target="_self" title="JORDAN3:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E10:37_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E11:37"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3_1_2_1_1_1"><i><font color="Green" title="E53">A24</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:37_3_1_2_1_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_3_1_2_1_1"><i><font color="Green" title="E52">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_2_1_2"><b>case </b></a><a NAME="E1:37_3_1_2_1_2"/><i><font color="Green" title="E52">A28</font></i>: 
<font color="Maroon" title="c8">n2</font> <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="c2">g</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_3_1_2_1_2_1"/><i><font color="Green" title="E53">A29</font></i>: 
<font color="Maroon" title="c7">n1</font> <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="c2">g</font>
 ;<br/><a NAME="E2:37_3_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:37_3_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E54">A30</font></i>: 
<font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><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="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:37_3_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n1</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="c3">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="c7">n1</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:37_3_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">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="c7">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E10:37"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3_1_2_1_2_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="jordan3.html#T27" target="_self" title="JORDAN3:th.27">Th27</a></i>;<br/><a NAME="E6:37_3_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E55">A31</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n1</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3_1"><i><font color="Green" title="E51">A22</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3_1_2_1_2"><i><font color="Green" title="E52">A28</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E7:37_3_1_2_1_2_1"/><i><font color="Green" title="E56">A32</font></i>: 
<font color="Maroon" title="c7">n1</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="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3_1_2_1_2_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:37_3_1_2_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">n1</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E3:37"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E6:37_3_1_2_1_2_1"><i><font color="Green" title="E55">A31</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:37_3_1_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n1</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jordan3.html#E6:37_3_1_2_1_2_1"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="jordan3.html#E7:37_3_1_2_1_2_1"><i><font color="Green" title="E56">A32</font></i></a>, <a class="ref" href="jordan3.html#T45" target="_self" title="JORDAN3:th.45">Th45</a></i>;<br/><b>hence </b><a NAME="E10:37_3_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan3.html#E7:37"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jordan3.html#E11:37"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3_1_2_1_2_1"><i><font color="Green" title="E53">A29</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:37_3_1_2_1_2"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_3_1_2_1_2"><i><font color="Green" title="E52">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_3_1_2_1_3"><b>case </b></a><b>that </b><a NAME="E1:37_3_1_2_1_3"/><i><font color="Green" title="E52">A33</font></i>: 
<font color="Maroon" title="c7">n1</font> <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="c2">g</font>
 <b>and </b><br/><a NAME="E2:37_3_1_2_1_3"/><i><font color="Green" title="E53">A34</font></i>: 
<font color="Maroon" title="c8">n2</font> <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="c2">g</font>
 ;<br/><div class="add"><a NAME="E3:37_3_1_2_1_3"/>
<font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3_1_2_1_3"><i><font color="Green" title="E52">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:37_3_1_2_1_3"/><b>then </b><i><font color="Green" title="E54">A35</font></i>: 
<font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><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="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:37_3_1_2_1_3"/><b>then </b><i><font color="Green" title="E55">A36</font></i>: 
<font color="Maroon" title="c7">n1</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="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><i><font color="Green" title="E56">A37</font></i>: <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n1</font> = 
<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="c3">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="c7">n1</font>
<b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E4:37_3_1_2_1_3"><i><font color="Green" title="E54">A35</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="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n1</font>
<b>by </b><i><a class="txt" href="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E4:37_3_1_2_1_3"><i><font color="Green" title="E54">A35</font></i></a>, <a class="ref" href="jordan3.html#T32" target="_self" title="JORDAN3:th.32">Th32</a></i>
;<br/><a NAME="E7:37_3_1_2_1_3"/>
<font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E2:37_3_1_2_1_3"><i><font color="Green" title="E53">A34</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E8:37_3_1_2_1_3"/><b>then </b><i><font color="Green" title="E57">A38</font></i>: 
<font color="Maroon" title="c8">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><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="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E18:37"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:37_3_1_2_1_3"/><b>then </b><i><font color="Green" title="E58">A39</font></i>: 
<font color="Maroon" title="c8">n2</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="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n2</font> = 
<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="c3">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="c8">n2</font>
<b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E8:37_3_1_2_1_3"><i><font color="Green" title="E57">A38</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="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n2</font>
<b>by </b><i><a class="txt" href="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E8:37_3_1_2_1_3"><i><font color="Green" title="E57">A38</font></i></a>, <a class="ref" href="jordan3.html#T32" target="_self" title="JORDAN3:th.32">Th32</a></i>
;<br/><b>hence </b><a NAME="E11:37_3_1_2_1_3"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E22:37"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="jordan3.html#E1:37_3"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="jordan3.html#E3:37_3"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="jordan3.html#E5:37_3_1_2_1_3"><i><font color="Green" title="E55">A36</font></i></a>, <a class="txt" href="jordan3.html#E6:37_3_1_2_1_3"><i><font color="Green" title="E56">A37</font></i></a>, <a class="txt" href="jordan3.html#E9:37_3_1_2_1_3"><i><font color="Green" title="E58">A39</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:37_3_1"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37_3"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E24:37"/><i><font color="Green" title="E49">A40</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E25:37"/><i><font color="Green" title="E50">A41</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <font color="Maroon" title="c2">g</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:37_4"/><i><font color="Green" title="E51">A42</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <font color="Maroon" title="c2">g</font> )
 ;<br/>

<a NAME="E2:37_4"/><b>then </b>
<font color="Maroon" title="c5">j</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="c2">g</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T65" title="BINARITH:th.65">BINARITH:65</a></i>;<br/>
<a NAME="E3:37_4"/><b>then </b><i><font color="Green" title="E52">A43</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">A42</font></i></a>, <a class="ref" href="jordan3.html#T51" target="_self" title="JORDAN3:th.51">Th51</a></i>;<br/>
<a NAME="E4:37_4"/><i><font color="Green" title="E53">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">A42</font></i></a></i>;<br/>
<a NAME="E5:37_4"/><i><font color="Green" title="E54">A45</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">A42</font></i></a>, <a class="ref" href="binarith.html#T66" title="BINARITH:th.66">BINARITH:66</a></i>;<br/>
<a NAME="E6:37_4"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E4:37_4"><i><font color="Green" title="E53">A44</font></i></a>, <a class="ref" href="jordan3.html#T51" target="_self" title="JORDAN3:th.51">Th51</a></i>;<br/>
<a NAME="E7:37_4"/><b>then </b><i><font color="Green" title="E55">A46</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E3:37_4"><i><font color="Green" title="E52">A43</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E8:37_4"/><i><font color="Green" title="E56">A47</font></i>: 
<font color="Maroon" title="c5">j</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="c2">g</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">A42</font></i></a>, <a class="ref" href="binarith.html#T65" title="BINARITH:th.65">BINARITH:65</a></i>;<br/>
<a NAME="E9:37_4"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">A42</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E10:37_4"/><b>then </b><i><font color="Green" title="E57">A48</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E11:37_4"/><i><font color="Green" title="E58">A49</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E5:37_4"><i><font color="Green" title="E54">A45</font></i></a>, <a class="txt" href="jordan3.html#E8:37_4"><i><font color="Green" title="E56">A47</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:37_4"/><i><font color="Green" title="E59">A50</font></i>: 
 <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</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="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan3.html#T41" target="_self" title="JORDAN3:th.41">Th41</a></i>;<br/>
<a NAME="E13:37_4"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="jordan3.html#E5:37_4"><i><font color="Green" title="E54">A45</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E14:37_4"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E15:37_4"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E10:37_4"><i><font color="Green" title="E57">A48</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:37_4"/><b>then </b><i><font color="Green" title="E60">A51</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_1"><b>now </b></a><div class="add"><a NAME="E1:37_4_1"/><i><font color="Green" title="E61">A52</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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> <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="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><a NAME="E2:37_4_1"/>
 <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</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="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan3.html#T41" target="_self" title="JORDAN3:th.41">Th41</a></i>;<br/><a NAME="E3:37_4_1"/><b>then </b><i><font color="Green" title="E62">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E1:37_4_1"><i><font color="Green" title="E61">A52</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_4_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan3.html#E3:37_4_1"><i><font color="Green" title="E62">A53</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="37_4_1_1_1_1"><b>case </b></a><a NAME="E1:37_4_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:37_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E1:37_4_1"><i><font color="Green" title="E61">A52</font></i></a>, <a class="ref" href="jordan3.html#T41" target="_self" title="JORDAN3:th.41">Th41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_4_1_1_1_2"><b>case </b></a><a NAME="E1:37_4_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:37_4_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:37_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E63">A54</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">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="E4:37_4_1_1_1_2"/><i><font color="Green" title="E64">A55</font></i>: 
<font color="Maroon" title="c5">j</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="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E4:37_4"><i><font color="Green" title="E53">A44</font></i></a>, <a class="txt" href="jordan3.html#E1:37_4_1"><i><font color="Green" title="E61">A52</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:37_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E65">A56</font></i>: 
<font color="Maroon" title="c5">j</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="c1">f</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E12:37_4"><i><font color="Green" title="E59">A50</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:37_4_1_1_1_2"/><i><font color="Green" title="E66">A57</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan3.html#E24:37"><i><font color="Green" title="E49">A40</font></i></a>, <a class="txt" href="jordan3.html#E1:37_4"><i><font color="Green" title="E51">A42</font></i></a>, <a class="txt" href="jordan3.html#E7:37_4"><i><font color="Green" title="E55">A46</font></i></a>, <a class="txt" href="jordan3.html#E3:37_4_1_1_1_2"><i><font color="Green" title="E63">A54</font></i></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></i>;<br/><a NAME="E7:37_4_1_1_1_2"/><i><font color="Green" title="E67">A58</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan3.html#E5:37_4"><i><font color="Green" title="E54">A45</font></i></a>, <a class="txt" href="jordan3.html#E5:37_4_1_1_1_2"><i><font color="Green" title="E65">A56</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<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="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E5:37_4"><i><font color="Green" title="E54">A45</font></i></a>, <a class="txt" href="jordan3.html#E4:37_4_1_1_1_2"><i><font color="Green" title="E64">A55</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="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan3.html#E8:37"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jordan3.html#E5:37_4"><i><font color="Green" title="E54">A45</font></i></a>, <a class="txt" href="jordan3.html#E4:37_4_1_1_1_2"><i><font color="Green" title="E64">A55</font></i></a>, <a class="ref" href="jordan3.html#T32" target="_self" title="JORDAN3:th.32">Th32</a></i>
;<br/><b>hence </b><a NAME="E9:37_4_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan3.html#E11:37_4"><i><font color="Green" title="E58">A49</font></i></a>, <a class="txt" href="jordan3.html#E16:37_4"><i><font color="Green" title="E60">A51</font></i></a>, <a class="txt" href="jordan3.html#E6:37_4_1_1_1_2"><i><font color="Green" title="E66">A57</font></i></a>, <a class="txt" href="jordan3.html#E7:37_4_1_1_1_2"><i><font color="Green" title="E67">A58</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:37_4_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:37_4"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E26:37"/><i><font color="Green" title="E51">A59</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E27:37"/><i><font color="Green" title="E52">A60</font></i>: 
for <font color="Olive" title="b1">j1</font>, <font color="Olive" title="b2">j2</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j1</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="Olive" title="b2">j2</font> holds <br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Olive" title="b1">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Olive" title="b2">j2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j1</font>, <font color="Maroon" title="c6">j2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:37_5"/><i><font color="Green" title="E53">A61</font></i>: 
<font color="Maroon" title="c5">j1</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="c6">j2</font>
 ;<br/>

<a NAME="E2:37_5"/>
<font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E3:37_5"/><b>then </b><i><font color="Green" title="E54">A62</font></i>: 
( <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_1_1"/>
( <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( ( <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 ) &amp; <font color="Maroon" title="c6">j2</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="c2">g</font> ) or <font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> )
 </span><b>by </b><i><a class="txt" href="jordan3.html#E3:37_5"><i><font color="Green" title="E54">A62</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="37_5_1_1_1"><b>case </b></a><a NAME="E1:37_5_1_1_1"/>
<font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:37_5_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:37_5_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:37_5_1_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_2"><b>case </b></a><b>that </b><a NAME="E1:37_5_1_1_2"/><i><font color="Green" title="E55">A63</font></i>: 
( <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <b>and </b><br/><a NAME="E2:37_5_1_1_2"/><i><font color="Green" title="E56">A64</font></i>: 
<font color="Maroon" title="c6">j2</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="c2">g</font>
 ;<br/><div class="add"><a NAME="E3:37_5_1_1_2"/>
<font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E2:37_5_1_1_2"><i><font color="Green" title="E56">A64</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:37_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">j1</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="c2">g</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_5"><i><font color="Green" title="E53">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:37_5_1_1_2"/><b>then </b><i><font color="Green" title="E57">A65</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j1</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E1:37_5_1_1_2"><i><font color="Green" title="E55">A63</font></i></a>, <a class="ref" href="jordan3.html#T51" target="_self" title="JORDAN3:th.51">Th51</a></i>;<br/><a NAME="E6:37_5_1_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_5_1_1_2"><i><font color="Green" title="E55">A63</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:37_5_1_1_2"/><b>then </b>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E1:37_5"><i><font color="Green" title="E53">A61</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:37_5_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:37_5_1_1_2"/><b>then </b><i><font color="Green" title="E58">A66</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E2:37_5_1_1_2"><i><font color="Green" title="E56">A64</font></i></a>, <a class="ref" href="jordan3.html#T51" target="_self" title="JORDAN3:th.51">Th51</a></i>;<br/><a NAME="E10:37_5_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E26:37"><i><font color="Green" title="E51">A59</font></i></a>, <a class="txt" href="jordan3.html#E1:37_5"><i><font color="Green" title="E53">A61</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/><a NAME="E11:37_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c6">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E12:37_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan3.html#E5:37_5_1_1_2"><i><font color="Green" title="E57">A65</font></i></a>, <a class="txt" href="jordan3.html#E9:37_5_1_1_2"><i><font color="Green" title="E58">A66</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><b>hence </b><a NAME="E13:37_5_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_3"><b>case </b></a><a NAME="E1:37_5_1_1_3"/>
<font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 ;<br/><div class="add"><a NAME="E2:37_5_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:37_5_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:37_5_1_1_3"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37_5"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c6">j2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E28:37"/><i><font color="Green" title="E53">A67</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E29:37"/>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</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="c2">g</font> &amp; not <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:37_6"/><i><font color="Green" title="E54">A68</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</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="c2">g</font> )
 ;<br/>

<div><i><font color="Green" title="E55">A69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6_1"><b>now </b></a><div class="add"><a NAME="E1:37_6_1"/>
<font color="Maroon" title="c5">j</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="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E12:37"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jordan3.html#E1:37_6"><i><font color="Green" title="E54">A68</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><a NAME="E2:37_6_1"/><b>then </b><i><font color="Green" title="E56">A70</font></i>: 
<font color="Maroon" title="c5">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="c3">p</font>,<font color="Maroon" title="c1">f</font>
 <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="E3:37_6_1"/>
 <a href="jordan3.html#K2" title="JORDAN3:func.2">Index</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">f</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="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan3.html#T41" target="_self" title="JORDAN3:th.41">Th41</a></i>;<br/><a NAME="E4:37_6_1"/><b>then </b>
<font color="Maroon" title="c5">j</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="jordan3.html#E2:37_6_1"><i><font color="Green" title="E56">A70</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:37_6_1"/>
<font color="Maroon" title="c5">j</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="c1">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:37_6"/><i><font color="Green" title="E56">A71</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j</font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E2:37"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jordan3.html#E4:37"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jordan3.html#E1:37_6"><i><font color="Green" title="E54">A68</font></i></a>, <a class="ref" href="jordan3.html#T51" target="_self" title="JORDAN3:th.51">Th51</a></i>;<br/>
<a NAME="E4:37_6"/><i><font color="Green" title="E57">A72</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_6"><i><font color="Green" title="E54">A68</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E5:37_6"/><i><font color="Green" title="E58">A73</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:37_6"><i><font color="Green" title="E54">A68</font></i></a>, <a class="txt" href="jordan3.html#E2:37_6"><i><font color="Green" title="E55">A69</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E6:37_6"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jordan3.html#E28:37"><i><font color="Green" title="E53">A67</font></i></a>, <a class="txt" href="jordan3.html#E1:37_6"><i><font color="Green" title="E54">A68</font></i></a>, <a class="txt" href="jordan3.html#E2:37_6"><i><font color="Green" title="E55">A69</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:37_6"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan3.html#E3:37_6"><i><font color="Green" title="E56">A71</font></i></a>, <a class="txt" href="jordan3.html#E4:37_6"><i><font color="Green" title="E57">A72</font></i></a>, <a class="txt" href="jordan3.html#E5:37_6"><i><font color="Green" title="E58">A73</font></i></a>, <a class="ref" href="jordan3.html#T36" target="_self" title="JORDAN3:th.36">Th36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:37"/><b>then </b>
( <font color="Maroon" title="c2">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c2">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c2">g</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c2">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> )
 
<b>by </b><i><a class="txt" href="jordan3.html#E20:37"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="jordan3.html#E23:37"><i><font color="Green" title="E48">A19</font></i></a>, <a class="txt" href="jordan3.html#E25:37"><i><font color="Green" title="E50">A41</font></i></a>, <a class="txt" href="jordan3.html#E27:37"><i><font color="Green" title="E52">A60</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/>
<a NAME="E31:37"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="topreal1.html#NR2" title="TOPREAL1:NR.2">is_S-Seq</a> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E32:37"/>
<font color="Maroon" title="c2">g</font> <a href="jordan3.html#R1" title="JORDAN3:pred.1">is_S-Seq_joining</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="jordan3.html#E17:37"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="jordan3.html#E19:37"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="jordan3.html#D3" target="_self" title="JORDAN3:def.3">Def3</a></i>;<br/>


</div>
