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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>, <font color="Maroon">r</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>, <font color="Maroon">CR</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K11">&lt;*</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_1.html#K11">*&gt;</a></span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">b2</font> = <font color="Maroon">b</font>, <font color="Maroon">a2</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<a NAME="E3:25"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">CR</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T60">FINSEQ_1:60</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<i><font color="Green">E37</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="finseq_1.html#K10">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
1 <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
3

;<br/>
<a NAME="E7:25"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<i><font color="Green">E39</font></i>: <font color="Maroon">f</font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T60">FINSEQ_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_8.html#K1">^</a> <font color="Maroon">CR</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E9:25"/>
 <a href="finseq_8.html#K3">ovlpart</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font>
 
<b>by </b><i>, <a class="ref" href="finseq_8.html#T14">FINSEQ_8:14</a></i>;<br/>
<a NAME="E10:25"/><b>then </b>
<font color="Maroon">CR</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K3">ovlpart</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="revrot_1.html#T2">REVROT_1:2</a></i>;<br/>
<a NAME="E11:25"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_8.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">CR</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_8.html#K3">ovlpart</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E12:25"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_8.html#K4">ovlcon</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_8.html#D3">FINSEQ_8:def 3</a></i>;<br/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#D1">JORDAN3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 1
<b>by </b><i><a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">CR</font>
<b>by </b><i>, <a class="ref" href="finseq_6.html#T6">FINSEQ_6:6</a></i>
;<br/>
<a NAME="E14:25"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 
<b>by </b><i>, <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a></i>;<br/>
<a NAME="E15:25"/><b>then </b><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 
<b>by </b><i><a class="ref" href="filerec1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_8.html#D11">FINSEQ_8:def 11</a></i>;<br/>
<a NAME="E16:25"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">CR</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r</font> <a href="finseq_8.html#R2">is_substring_of</a> <font color="Maroon">f</font>,1 )
 
<b>by </b><i>, <a class="ref" href="finseq_8.html#T19">FINSEQ_8:19</a>, </i>;<br/>
<a NAME="E17:25"/><b>then </b><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">CR</font> <a href="finseq_8.html#K1">^</a> <font color="Maroon">r</font> <a href="finseq_8.html#R2">is_substring_of</a>  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>,1 )
 
<b>by </b><i><a class="ref" href="filerec1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_8.html#D11">FINSEQ_8:def 11</a></i>;<br/>
<i><font color="Green">E55</font></i>: <font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default">2 <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>

<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="rfinseq.html#K1">/^</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon">CR</font>
<b>by </b><i>, <a class="ref" href="finseq_6.html#T50">FINSEQ_6:50</a></i>
;<br/>
<i><font color="Green">E56</font></i>: <font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> 0
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
;<br/>
<a NAME="E20:25"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E21:25"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E22:25"/><b>then </b><i><font color="Green">E58</font></i>: 
not <font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 
<b>by </b><i>, , , <a class="ref" href="finseq_8.html#T21">FINSEQ_8:21</a></i>;<br/>
<a NAME="E23:25"/><i><font color="Green">E59</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; <font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default">2 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E24:25"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25_6"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">r</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 ;<br/>

<a NAME="E2:25_6"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="filerec1.html#T2" target="_self">Th2</a>, <a class="ref" href="filerec1.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:25_6"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:25_6"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E25:25"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_8.html#K7">instr</a> 1,<font color="Maroon">r</font>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a>, </i>;<br/>
<a NAME="E26:25"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E27:25"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">CR</font> <a href="finseq_8.html#R5">is_terminated_by</a> <font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="finseq_8.html#T28">FINSEQ_8:28</a></i>;<br/>
<a NAME="E28:25"/>
<font color="Maroon">r</font> <a href="finseq_8.html#R5">is_terminated_by</a> <font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="filerec1.html#T3" target="_self">Th3</a>, , , <a class="ref" href="filerec1.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_8.html#D12">FINSEQ_8:def 12</a></i>;<br/>
<b>hence </b><a NAME="E29:25"/>
( <font color="Maroon">CR</font> <a href="filerec1.html#R1" target="_self">is_a_record_of</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> &amp; <font color="Maroon">r</font> <a href="filerec1.html#R1" target="_self">is_a_record_of</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> )
 <b>by </b><i>, , , </i>;<br/>


</div>
