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

<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">r</font>, <font color="Olive" title="b5">CR</font> being    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b5">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="Olive" title="b2">b</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Olive" title="b3">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="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> &amp; <font color="Olive" title="b4">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> holds <br/>( <font color="Olive" title="b5">CR</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b5">CR</font> &amp; <font color="Olive" title="b4">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b5">CR</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">CR</font> being    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>  st <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="Olive" title="b3">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="Olive" title="b1">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="Olive" title="b2">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> holds <br/>( <font color="Olive" title="b3">CR</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">CR</font> &amp; <font color="Olive" title="b2">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">CR</font> )</span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="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> implies ( <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> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:25"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:25"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:25"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:25"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="filerec1.html#T10" target="_self" onmouseover="rs('filerec1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E6:25"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
( <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> &amp;  <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 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="finseq_1.html#T43" onmouseover="rs('finseq_1/T43')" onmouseout="rh()">FINSEQ_1:43</a></span>
<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_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">CR</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="filerec1.html#T14" target="_self" onmouseover="rs('filerec1/T14')" onmouseout="rh()">Th14</a></span>
;<br/>
<a NAME="E8:25"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#T14" onmouseover="rs('finseq_8/T14')" onmouseout="rh()">FINSEQ_8:14</a></span>;<br/>
<a NAME="E9:25"/><span class="kw">then </span>
<font color="Maroon" title="c6">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ: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">(<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T167" onmouseover="rs('finseq_6/T167')" onmouseout="rh()">FINSEQ_6:167</a></span>;<br/>
<a NAME="E10:25"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/>
<a NAME="E11:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D3" onmouseover="rs('finseq_8/D3')" onmouseout="rh()">FINSEQ_8:def 3</a></span>;<br/>
<a NAME="E12:25"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<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_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="finseq_1.html#T43" onmouseover="rs('finseq_1/T43')" onmouseout="rh()">FINSEQ_1:43</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E20">A8</font></span>:  <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
<br/>.= 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>
<br/>.= 
3

;<br/>
<a NAME="E14:25"/><span class="kw">then </span>
<font color="Maroon" title="c6">CR</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="finseq_8.html#T19" onmouseover="rs('finseq_8/T19')" onmouseout="rh()">FINSEQ_8:19</a></span>;<br/>
<a NAME="E15:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c6">CR</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="finseq_8.html#D11" onmouseover="rs('finseq_8/D11')" onmouseout="rh()">FINSEQ_8:def 11</a></span>;<br/>
<a NAME="E16:25"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/>
<span class="kw">then </span> <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">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#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#D3" onmouseover="rs('finseq_6/D3')" onmouseout="rh()">FINSEQ_6:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T8" onmouseover="rs('nat_2/T8')" onmouseout="rh()">NAT_2:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1

<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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="finseq_1.html#D16" onmouseover="rs('finseq_1/D16')" onmouseout="rh()">FINSEQ_1:def 16</a></span>
<br/>.= 
<font color="Maroon" title="c6">CR</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="finseq_6.html#T4" onmouseover="rs('finseq_6/T4')" onmouseout="rh()">FINSEQ_6:4</a></span>
;<br/>
<a NAME="E18:25"/><span class="kw">then </span>
<font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/>
<a NAME="E19:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="finseq_8.html#D11" onmouseover="rs('finseq_8/D11')" onmouseout="rh()">FINSEQ_8:def 11</a></span>;<br/>
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>

<br/>.= 
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
<br/>.= 
<font color="Maroon" title="c5">r</font>

;<br/>
<a NAME="E21:25"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E16">A4</font></span></a></span>;<br/>
<a NAME="E22:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A12</font></span>: 
 not <font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="finseq_8.html#T21" onmouseover="rs('finseq_8/T21')" onmouseout="rh()">FINSEQ_8:21</a></span>;<br/>
<a NAME="E23:25"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
 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#NK1" title="NUMBERS:NK.1">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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> 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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> implies <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:25_5"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:25_5"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:25_5"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2</span><br/>

<a NAME="E4:25_5"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E1:25_5"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E3:25_5"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E5:25_5"/><span class="kw">then </span>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:25_5"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c5">r</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</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="nat_d.html#K7" title="NAT_D:func.7">-'</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#K2" title="RFINSEQ:func.2">/^</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
<br/>.= 
<font color="Maroon" title="c6">CR</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="finseq_6.html#T46" onmouseover="rs('finseq_6/T46')" onmouseout="rh()">FINSEQ_6:46</a></span>
;<br/>
<a NAME="E25:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A16</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:25"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="finseq_8.html#D10" onmouseover="rs('finseq_8/D10')" onmouseout="rh()">FINSEQ_8:def 10</a></span>;<br/>
<a NAME="E26:25"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E27:25"/><span class="kw">then </span>
<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="nat_d.html#K7" title="NAT_D:func.7">-'</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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/>
<a NAME="E28:25"/><span class="kw">then </span>
( <font color="Maroon" title="c6">CR</font> <a href="finseq_8.html#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Maroon" title="c6">CR</font> &amp; <font color="Maroon" title="c5">r</font> <a href="finseq_8.html#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Maroon" title="c6">CR</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E25:25"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="finseq_8.html#T28" onmouseover="rs('finseq_8/T28')" onmouseout="rh()">FINSEQ_8:28</a>, <a class="ref" href="finseq_8.html#D12" onmouseover="rs('finseq_8/D12')" onmouseout="rh()">FINSEQ_8:def 12</a></span>;<br/>
<span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:25"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E15:25"><span class="lab"><font color="Green" title="E21">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
