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<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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">CR</font>, <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font>, <font color="Olive" title="b4">f</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="b2">r1</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Olive" title="b1">CR</font> &amp; <font color="Olive" title="b3">r2</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Olive" title="b1">CR</font> &amp; <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Olive" title="b3">r2</font> holds <br/>( <font color="Olive" title="b2">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b1">CR</font> &amp; <font color="Olive" title="b3">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b1">CR</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">CR</font>, <font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font>, <font color="Maroon" title="c5">f</font> be    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font> implies ( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c2">CR</font>
 <span class="kw">and </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:47_1"/>
( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><span class="kw">suppose </span></a><a NAME="E1:47_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )</span><br/><div class="add"><a NAME="E2:47_1_1"/><span class="kw">then </span>
(  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  &amp;  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_1_1"/>
( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><span class="kw">suppose </span></a><a NAME="E1:47_1_2"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )</span><br/><div class="add"><a NAME="E2:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/><a NAME="E3:47_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E4:47_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="filerec1.html#T3" target="_self" title="FILEREC1:th.3">Th3</a></span>;<br/><a NAME="E5:47_1_2"/><span class="kw">then </span>
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="filerec1.html#T29" target="_self" title="FILEREC1:th.29">Th29</a></span>;<br/><span class="kw">hence </span><a NAME="E6:47_1_2"/>
( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E2:47_1_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E4:47_1_2"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3"><span class="kw">suppose </span></a><a NAME="E1:47_1_3"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )</span><br/><div class="add"><a NAME="E2:47_1_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E3:47_1_3"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">r1</font></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> <font color="Maroon" title="c4">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_3"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:47_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A8</font></span>: 
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="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E5:47_1_3"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_3"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A9</font></span>:  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlcon</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D3" title="FINSEQ_8:def.3">FINSEQ_8:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
<br/>.= 
<font color="Maroon" title="c3">r1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></span>
<br/>.= 
<font color="Maroon" title="c3">r1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>
;<br/><a NAME="E7:47_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filerec1.html#T31" target="_self" title="FILEREC1:th.31">Th31</a></span>;<br/><a NAME="E8:47_1_3"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( 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">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_3"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E6:47_1_3"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E9:47_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A10</font></span>: 
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/><a NAME="E10:47_1_3"/>
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_3"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:47_1_3"/>
( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E9:47_1_3"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_4"><span class="kw">suppose </span></a><a NAME="E1:47_1_4"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )</span><br/><div class="add"><a NAME="E2:47_1_4"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">r1</font></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> <font color="Maroon" title="c4">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:47_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A12</font></span>: 
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="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E4:47_1_4"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_4"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A13</font></span>:  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlcon</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D3" title="FINSEQ_8:def.3">FINSEQ_8:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p5">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
<br/>.= 
<font color="Maroon" title="c3">r1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></span>
<br/>.= 
<font color="Maroon" title="c3">r1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></span>
;<br/><a NAME="E6:47_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filerec1.html#T31" target="_self" title="FILEREC1:th.31">Th31</a></span>;<br/><a NAME="E7:47_1_4"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( 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">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_4"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E5:47_1_4"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:47_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/><a NAME="E9:47_1_4"/>
<font color="Maroon" title="c2">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <font color="Maroon" title="c4">r2</font></span>)</span>,<font color="Maroon" title="c2">CR</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="filerec1.html#T37" target="_self" title="FILEREC1:th.37">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E10:47_1_4"/>
( <font color="Maroon" title="c3">r1</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c2">CR</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E8:47_1_4"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
