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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E27">A1</font></i>: 
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_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="46_1_1"><b>suppose </b></a><a NAME="E1:46_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
(  <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:()"><i><font color="Red">::  thesis: </font></i></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="E2:46_1_1"/><b>then </b><i><font color="Green" title="E29">A3</font></i>: 
 <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> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <b>by </b><i><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/><a NAME="E3:46_1_1"/>
 <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> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/><b>hence </b><a NAME="E4:46_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> )
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="filerec1.html#E2:46_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/><i><font color="Green" title="E28">A4</font></i>: 
(  <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:()"><i><font color="Red">::  thesis: </font></i></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="E2:46_1_2"/><b>then </b><i><font color="Green" title="E29">A5</font></i>: 
 <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> 
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E3:46_1_2"/><i><font color="Green" title="E30">A6</font></i>: 
 <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> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46_1_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/><a NAME="E4:46_1_2"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r2</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="filerec1.html#E2:46_1_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="filerec1.html#T3" target="_self" title="FILEREC1:th.3">Th3</a></i>;<br/><a NAME="E5:46_1_2"/><b>then </b>
 <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> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="filerec1.html#T29" target="_self" title="FILEREC1:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E6:46_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> )
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="filerec1.html#E3:46_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="filerec1.html#E4:46_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3"><b>suppose </b></a><a NAME="E1:46_1_3"/><i><font color="Green" title="E28">A8</font></i>: 
(  <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:()"><i><font color="Red">::  thesis: </font></i></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="E2:46_1_3"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:46_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>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E4:46_1_3"/><b>then </b>
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>
 <b>by </b><i><a class="txt" href="filerec1.html#E2:46_1_3"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:46_1_3"/><b>then </b><i><font color="Green" title="E30">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:46_1_3"/><i><font color="Green" title="E31">A11</font></i>: 
 <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>
 <b>by </b><i><a class="ref" href="filerec1.html#T31" target="_self" title="FILEREC1:th.31">Th31</a></i>;<br/> <a href="jordan3.html#K1" title="JORDAN3:func.1">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>
<b>by </b><i><a class="txt" href="filerec1.html#E2:46_1_3"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
<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>
<b>by </b><i><a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></i>
<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#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>
<b>by </b><i><a class="ref" href="finseq_8.html#D3" title="FINSEQ_8:def.3">FINSEQ_8:def 3</a></i>
<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#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>
<b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<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>
<b>by </b><i><a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></i>
<br/>.= 
<font color="Maroon" title="c3">r1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
;<br/><a NAME="E8:46_1_3"/><b>then </b>
(  <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#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="jordan3.html#K1" title="JORDAN3:func.1">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> ) )
 <b>by </b><i><a class="txt" href="filerec1.html#E5:46_1_3"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="filerec1.html#E6:46_1_3"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:46_1_3"/><b>then </b><i><font color="Green" title="E32">A12</font></i>: 
 <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> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <b>by </b><i><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/><a NAME="E10:46_1_3"/>
 <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> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46_1_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/><b>hence </b><a NAME="E11:46_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> )
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="filerec1.html#E9:46_1_3"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_4"><b>suppose </b></a><a NAME="E1:46_1_4"/>
(  <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:()"><i><font color="Red">::  thesis: </font></i></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="E2:46_1_4"/><b>then </b><i><font color="Green" title="E28">A13</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:46_1_4"/>
 <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>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E4:46_1_4"/><b>then </b>
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>
 <b>by </b><i><a class="txt" href="filerec1.html#E2:46_1_4"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E5:46_1_4"/><b>then </b><i><font color="Green" title="E29">A14</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:46_1_4"/><i><font color="Green" title="E30">A15</font></i>: 
 <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>
 <b>by </b><i><a class="ref" href="filerec1.html#T31" target="_self" title="FILEREC1:th.31">Th31</a></i>;<br/> <a href="jordan3.html#K1" title="JORDAN3:func.1">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>
<b>by </b><i><a class="txt" href="filerec1.html#E2:46_1_4"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
<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>
<b>by </b><i><a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></i>
<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#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>
<b>by </b><i><a class="ref" href="finseq_8.html#D3" title="FINSEQ_8:def.3">FINSEQ_8:def 3</a></i>
<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#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>
<b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<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>
<b>by </b><i><a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></i>
<br/>.= 
<font color="Maroon" title="c3">r1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
;<br/><a NAME="E8:46_1_4"/><b>then </b>
(  <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#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="jordan3.html#K1" title="JORDAN3:func.1">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> ) )
 <b>by </b><i><a class="txt" href="filerec1.html#E5:46_1_4"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="filerec1.html#E6:46_1_4"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:46_1_4"/><b>then </b><i><font color="Green" title="E31">A16</font></i>: 
 <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> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> 
 <b>by </b><i><a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></i>;<br/><a NAME="E10:46_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#K8" title="FINSEQ_8:func.8">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
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="filerec1.html#T37" target="_self" title="FILEREC1:th.37">Th37</a></i>;<br/><b>hence </b><a NAME="E11:46_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> )
 <b>by </b><i><a class="txt" href="filerec1.html#E1:46"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="filerec1.html#E9:46_1_4"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="filerec1.html#D1" target="_self" title="FILEREC1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
