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<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">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#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Olive" title="b1">CR</font> &amp; <font color="Olive" title="b3">r2</font> <a href="finseq_8.html#R4" title="FINSEQ_8:pred.4">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_1.html#K8" title="FINSEQ_1:func.8">^</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" 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="c3">r1</font> <a href="finseq_8.html#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Maroon" title="c2">CR</font> &amp; <font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R4" title="FINSEQ_8:pred.4">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_1.html#K8" title="FINSEQ_1:func.8">^</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="E1:44"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c3">r1</font> <a href="finseq_8.html#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Maroon" title="c2">CR</font>
 <span class="kw">and </span><br/><a NAME="E2:44"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c4">r2</font> <a href="finseq_8.html#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Maroon" title="c2">CR</font>
 <span class="kw">and </span><br/><a NAME="E3:44"/><span class="lab"><font color="Green" title="E28">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_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">r2</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="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="44_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:44_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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1"><span class="kw">suppose </span></a><a NAME="E1:44_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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</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="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="E2:44_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">r1</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="c5">f</font>,<font color="Maroon" title="c2">CR</font>) &amp; <font color="Maroon" title="c4">r2</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="c5">f</font>,<font color="Maroon" title="c2">CR</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:44_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="#E1:44"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E27">A2</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2"><span class="kw">suppose </span></a><a NAME="E1:44_1_2"/><span class="lab"><font color="Green" title="E29">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</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="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="E2:44_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c3">r1</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="c5">f</font>,<font color="Maroon" title="c2">CR</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><a NAME="E3:44_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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>;<br/><a NAME="E4:44_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">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="#E3:44"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="filerec1.html#T3" target="_self" onmouseover="rs('filerec1/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E5:44_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</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="c5">f</font>,<font color="Maroon" title="c2">CR</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="filerec1.html#T28" target="_self" onmouseover="rs('filerec1/T28')" onmouseout="rh()">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E6:44_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="#E1:44"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E2:44_1_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E4:44_1_2"><span class="lab"><font color="Green" title="E31">A6</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_3"><span class="kw">suppose </span></a><a NAME="E1:44_1_3"/><span class="lab"><font color="Green" title="E29">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</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="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="E2:44_1_3"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#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="c4">r2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_3"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:44_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">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="#E3:44"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E4:44_1_3"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_3"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">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#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">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" onmouseover="rs('finseq_8/D11')" onmouseout="rh()">FINSEQ_8:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K3" title="FINSEQ_8:func.3">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" onmouseover="rs('finseq_8/D3')" onmouseout="rh()">FINSEQ_8:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K3" title="FINSEQ_8:func.3">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="#E3:44"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</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#T22" onmouseover="rs('finseq_5/T22')" onmouseout="rh()">FINSEQ_5:22</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#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>
;<br/><a NAME="E6:44_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#K8" title="FINSEQ_8:func.8">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#T30" target="_self" onmouseover="rs('filerec1/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E7:44_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#K5" title="NUMBERS:func.5">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#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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="#E3:44_1_3"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E5:44_1_3"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:44_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c3">r1</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="c5">f</font>,<font color="Maroon" title="c2">CR</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><a NAME="E9:44_1_3"/>
<font color="Maroon" title="c4">r2</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="c5">f</font>,<font color="Maroon" title="c2">CR</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_3"><span class="lab"><font color="Green" title="E29">A7</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/><span class="kw">hence </span><a NAME="E10:44_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="#E1:44"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E8:44_1_3"><span class="lab"><font color="Green" title="E32">A10</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_4"><span class="kw">suppose </span></a><a NAME="E1:44_1_4"/><span class="lab"><font color="Green" title="E29">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#K5" title="NUMBERS:func.5">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#K5" title="NUMBERS:func.5">0</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="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="E2:44_1_4"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#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="c4">r2</font></span>)</span>
 <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/><a NAME="E3:44_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">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="#E3:44"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E4:44_1_4"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_4"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">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#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">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" onmouseover="rs('finseq_8/D11')" onmouseout="rh()">FINSEQ_8:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K3" title="FINSEQ_8:func.3">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" onmouseover="rs('finseq_8/D3')" onmouseout="rh()">FINSEQ_8:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K3" title="FINSEQ_8:func.3">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="#E3:44"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</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#T22" onmouseover="rs('finseq_5/T22')" onmouseout="rh()">FINSEQ_5:22</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#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>
;<br/><a NAME="E6:44_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#K8" title="FINSEQ_8:func.8">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#T30" target="_self" onmouseover="rs('filerec1/T30')" onmouseout="rh()">Th30</a></span>;<br/><a NAME="E7:44_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#K5" title="NUMBERS:func.5">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#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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="#E3:44_1_4"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E5:44_1_4"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:44_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A14</font></span>: 
<font color="Maroon" title="c3">r1</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="c5">f</font>,<font color="Maroon" title="c2">CR</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><a NAME="E9:44_1_4"/>
<font color="Maroon" title="c2">CR</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K8" title="FINSEQ_8:func.8">addcr</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">r1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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="#E1:44"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="filerec1.html#T36" target="_self" onmouseover="rs('filerec1/T36')" onmouseout="rh()">Th36</a></span>;<br/><span class="kw">hence </span><a NAME="E10:44_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="#E1:44"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E8:44_1_4"><span class="lab"><font color="Green" title="E32">A14</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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