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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">CR</font> be    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c6">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">b2</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c8">a2</font> = <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="filerec1.html#T10" target="_self" title="FILEREC1:th.10">Th10</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c6">CR</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="c3">b</font>
 
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green" title="E17">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<i><font color="Green" title="E18">A5</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">CR</font></span>)</span> <a href="nat_1.html#K1" title="NAT_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="c5">r</font></span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E4:25"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
3

;<br/>
<a NAME="E7:25"/><i><font color="Green" title="E19">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<i><font color="Green" title="E20">A7</font></i>: <font color="Maroon" title="c4">f</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#T60" title="FINSEQ_1:th.60">FINSEQ_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c6">CR</font>
<b>by </b><i><a class="ref" href="filerec1.html#T14" target="_self" title="FILEREC1:th.14">Th14</a>, <a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a></i>
;<br/>
<a NAME="E9:25"/>
 <a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">CR</font>
 
<b>by </b><i><a class="txt" href="filerec1.html#E8:25"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="finseq_8.html#T14" title="FINSEQ_8:th.14">FINSEQ_8:14</a></i>;<br/>
<a NAME="E10:25"/><b>then </b>
<font color="Maroon" title="c6">CR</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.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="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font></span>)</span></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="revrot_1.html#T2" title="REVROT_1:th.2">REVROT_1:2</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">CR</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="finseq_8.html#K3" title="FINSEQ_8:func.3">ovlpart</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E12:25"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
 <a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlcon</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="finseq_8.html#D3" title="FINSEQ_8:def.3">FINSEQ_8:def 3</a></i>;<br/>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">CR</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E5:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="jordan3.html#D1" title="JORDAN3:def.1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 0</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
<b>by </b><i><a class="ref" href="finseq_5.html#T31" title="FINSEQ_5:th.31">FINSEQ_5:31</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">b2</font>,<font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c7">b2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon" title="c6">CR</font>
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_6.html#T6" title="FINSEQ_6:th.6">FINSEQ_6:6</a></i>
;<br/>
<a NAME="E14:25"/><b>then </b>
( 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="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  )
 
<b>by </b><i><a class="txt" href="filerec1.html#E6:25"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/>
<a NAME="E15:25"/><b>then </b><i><font color="Green" title="E22">A9</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="c4">f</font> &amp;  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">addcr</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  )
 
<b>by </b><i><a class="txt" href="filerec1.html#E12:25"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></i>;<br/>
<a NAME="E16:25"/>
( 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="c4">f</font> &amp; <font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c5">r</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <font color="Maroon" title="c4">f</font>,1 )
 
<b>by </b><i><a class="txt" href="filerec1.html#E4:25"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="finseq_8.html#T19" title="FINSEQ_8:th.19">FINSEQ_8:19</a>, <a class="txt" href="filerec1.html#E6:25"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/>
<a NAME="E17:25"/><b>then </b><i><font color="Green" title="E23">A10</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="c4">f</font> &amp; <font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c5">r</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">addcr</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font>,1 )
 
<b>by </b><i><a class="txt" href="filerec1.html#E12:25"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></i>;<br/>
<i><font color="Green" title="E24">A11</font></i>: <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default">2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>

<br/>.= 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c7">b2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 1
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon" title="c6">CR</font>
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_6.html#T50" title="FINSEQ_6:th.50">FINSEQ_6:50</a></i>
;<br/>
<i><font color="Green" title="E25">A12</font></i>: <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> 0
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon" title="c5">r</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T31" title="FINSEQ_5:th.31">FINSEQ_5:31</a></i>
;<br/>
<a NAME="E20:25"/><i><font color="Green" title="E26">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E21:25"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="filerec1.html#E19:25"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E22:25"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
not <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 
<b>by </b><i><a class="txt" href="filerec1.html#E20:25"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="filerec1.html#E3:25"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="filerec1.html#E1:25"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="finseq_8.html#T21" title="FINSEQ_8:th.21">FINSEQ_8:21</a></i>;<br/>
<a NAME="E23:25"/><i><font color="Green" title="E28">A15</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default">2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  )
 
<b>by </b><i><a class="txt" href="filerec1.html#E18:25"><i><font color="Green" title="E24">A11</font></i></a></i>;<br/>
<a NAME="E24:25"/>
for <font color="Olive" title="b1">j</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 <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</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="E1:25_6"/><i><font color="Green" title="E29">A16</font></i>: 
( <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  )
 ;<br/>

<a NAME="E2:25_6"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="filerec1.html#E22:25"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="filerec1.html#E1:25_6"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:25_6"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E4:25_6"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E25:25"/><b>then </b><i><font color="Green" title="E29">A17</font></i>: 
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> 1,<font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c6">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a>, <a class="txt" href="filerec1.html#E23:25"><i><font color="Green" title="E28">A15</font></i></a></i>;<br/>
<a NAME="E26:25"/><i><font color="Green" title="E30">A18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="filerec1.html#E5:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="filerec1.html#E7:25"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E27:25"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c6">CR</font>
 
<b>by </b><i><a class="ref" href="finseq_8.html#T28" title="FINSEQ_8:th.28">FINSEQ_8:28</a></i>;<br/>
<a NAME="E28:25"/>
<font color="Maroon" title="c5">r</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c6">CR</font>
 
<b>by </b><i><a class="txt" href="filerec1.html#E25:25"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="filerec1.html#E5:25"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="filerec1.html#E7:25"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="filerec1.html#E26:25"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="finseq_8.html#D12" title="FINSEQ_8:def.12">FINSEQ_8:def 12</a></i>;<br/>
<b>hence </b><a NAME="E29:25"/>
( <font color="Maroon" title="c6">CR</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font> &amp; <font color="Maroon" title="c5">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">CR</font> )
 <b>by </b><i><a class="txt" href="filerec1.html#E27:25"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="filerec1.html#E15:25"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="filerec1.html#E17:25"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></i>;<br/>


</div>
