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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#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">f</font>, <font color="Olive" title="b2">CR</font> being   <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>  ex <font color="Olive" title="b3">r</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="b3">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">CR</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">CR</font> be    <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being   <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">i0</font> =  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font>;<br/>
<a NAME="E1:45"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_1"/>
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/>

<a NAME="E2:45_1"/><span class="kw">then </span>
not <font color="Maroon" title="c3">CR</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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>,1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:45_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filerec1.html#T41" target="_self" title="FILEREC1:th.41">Th41</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>
<a NAME="E2:45"/><span class="kw">then </span>
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/>
<a NAME="E3:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<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="E4:45"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E5:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2"><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="E1:45_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="45_2_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  ex <font color="Olive" title="b1">r</font> being   <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span><br/><div class="add"><a NAME="E2:45_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:45_2_1"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filerec1.html#T23" target="_self" title="FILEREC1:th.23">Th23</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="45_2_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_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="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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">  ex <font color="Olive" title="b1">r</font> being   <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">r</font> = <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/><a NAME="E2:45_2_2"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="#E1:45"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_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="E1:45_2_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</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">CR</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="45_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</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"> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span><br/><div class="add"><a NAME="E2:45_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c3">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</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="E3:45_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T86" title="FINSEQ_6:th.86">FINSEQ_6:86</a></span>;<br/><a NAME="E4:45_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">CR</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="finseq_8.html#K4" title="FINSEQ_8:func.4">ovlpart</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><a NAME="E5:45_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <a href="finseq_8.html#K5" title="FINSEQ_8:func.5">ovlcon</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <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/><a NAME="E6:45_2_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E7:45_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E8:45_2_2_1_1_1"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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> <font color="Maroon" title="c2">f</font> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_2_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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"> ( 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="c2">f</font> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/>

<a NAME="E2:45_2_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_1_1_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="E1:45_2_2_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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="c2">f</font> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<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/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="finseq_5.html#T80" title="FINSEQ_5:th.80">FINSEQ_5:80</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>
<br/>.= 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:45_2_2_1_1_1_1_1_1"/>
( 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="c2">f</font> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></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="45_2_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 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="c2">f</font> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )</span><br/><div class="add"><a NAME="E2:45_2_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T19" title="FINSEQ_5:th.19">FINSEQ_5:19</a></span>;<br/> <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">f</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <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#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:45_2_2_1_1_1_1_1_2"/>
( 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="c2">f</font> &amp;  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <font color="Maroon" title="c2">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E1:45_2_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E2:45_2_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><span class="kw">end;</span></div><a NAME="E9:45_2_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="finseq_8.html#D12" title="FINSEQ_8:def.12">FINSEQ_8:def 12</a></span>;<br/><a NAME="E10:45_2_2_1_1_1"/>
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="finseq_8.html#D11" title="FINSEQ_8:def.11">FINSEQ_8:def 11</a></span>;<br/><a NAME="E11:45_2_2_1_1_1"/><span class="kw">then </span>
 <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">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="#E8:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E36">A10</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="E12:45_2_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2_2_1_1_1"><span class="lab"><font color="Green" title="E37">A15</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="45_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_2"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</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"> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span><br/><div class="add"><a NAME="E2:45_2_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="#E1:45"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/><a NAME="E3:45_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</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"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E4:45_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/><a NAME="E5:45_2_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/><a NAME="E6:45_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E7:45_2_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:45_2_2_1_1_2"/>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/><a NAME="E9:45_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E10:45_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</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"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T18" title="FINSEQ_5:th.18">FINSEQ_5:18</a></span>;<br/><a NAME="E11:45_2_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T28" title="FINSEQ_5:th.28">FINSEQ_5:28</a></span>;<br/><div><span class="lab"><font color="Green" title="E38">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_2_1_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="E1:45_2_2_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span><br/><div class="add"><a NAME="E2:45_2_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</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/><span class="kw">hence </span><a NAME="E3:45_2_2_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="txt" href="#E11:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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="45_2_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span><br/><div class="add"><a NAME="E2:45_2_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E3:45_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E4:45_2_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:45_2_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E3:45_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</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><span class="kw">end;</span></div><a NAME="E13:45_2_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E14:45_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E6:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E15:45_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E16:45_2_2_1_1_2"/>
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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"><span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> ) ) ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  not <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:45_2_2_1_1_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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"><span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  not <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1 )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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"><span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> ) )</span><br/>

<a NAME="E2:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</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/>
<span class="kw">hence </span><a NAME="E3:45_2_2_1_1_2_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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"><span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">f2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<a NAME="E4:45_2_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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="c3">CR</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E5:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E6:45_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E7:45_2_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E8:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E9:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E15:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E4:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E44">A30</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E6:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font>

;<br/>
<a NAME="E11:45_2_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E12:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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="c3">CR</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E13:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T42" title="NAT_D:th.42">NAT_D:42</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E11:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font>

;<br/>
<a NAME="E15:45_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A32</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T83" title="FINSEQ_5:th.83">FINSEQ_5:83</a></span>;<br/>
<a NAME="E16:45_2_2_1_1_2_2_1"/>
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/>
<a NAME="E17:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E15:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E18:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</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="E19:45_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A33</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E20:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E48">A34</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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="c3">CR</font></span>)</span>

;<br/>
<a NAME="E22:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E23:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E24:45_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A35</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E25:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">CR</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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="E26:45_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A36</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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="c3">CR</font></span>)</span>

;<br/>
<a NAME="E28:45_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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"><span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E6:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E26:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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"><span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E10:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E19:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>; <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">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span><br/>

<a NAME="E30:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">CR</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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="E31:45_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A37</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E32:45_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E33:45_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E34:45_2_2_1_1_2_2_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2_1_1_2_2_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="c7">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c7">j</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; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a>  implies <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E53">A39</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E2:45_2_2_1_1_2_2_1_5"/>
<font color="Maroon" title="c7">j</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 class="kw">and </span><br/><a NAME="E3:45_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E54">A40</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</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="c7">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E55">A41</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T83" title="FINSEQ_5:th.83">FINSEQ_5:83</a></span>
;<br/>
<a NAME="E5:45_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E56">A42</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/>
<a NAME="E6:45_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E57">A43</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2_1_1_2_2_1_5_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:45_2_2_1_1_2_2_1_5_2"/><span class="lab"><font color="Green" title="E58">A44</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E6:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E21:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E31:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E3:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="ref" href="nat_d.html#T56" title="NAT_D:th.56">NAT_D:56</a></span>;<br/><a NAME="E4:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A45</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filerec1.html#T4" target="_self" title="FILEREC1:th.4">Th4</a></span>;<br/><a NAME="E5:45_2_2_1_1_2_2_1_5_2"/><span class="lab"><font color="Green" title="E60">A46</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A47</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 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">CR</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E57">A43</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
;<br/><a NAME="E7:45_2_2_1_1_2_2_1_5_2"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E1:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E8:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">j</font> <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="c3">CR</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E31:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E9:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E10:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="txt" href="#E5:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T83" title="FINSEQ_5:th.83">FINSEQ_5:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E10:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="ref" href="finseq_5.html#T80" title="FINSEQ_5:th.80">FINSEQ_5:80</a></span>
<br/>.= 
<font color="Maroon" title="c3">CR</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E3:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="txt" href="#E5:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E4:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="txt" href="#E6:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>
;<br/><a NAME="E12:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A49</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a></span>;<br/><a NAME="E13:45_2_2_1_1_2_2_1_5_2"/>
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"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="finseq_8.html#D8" title="FINSEQ_8:def.8">FINSEQ_8:def 8</a></span>;<br/><a NAME="E14:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:45_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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="#E12:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E63">A49</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="E16:45_2_2_1_1_2_2_1_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E1:45_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="txt" href="#E1:45_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</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>
<span class="kw">hence </span><a NAME="E8:45_2_2_1_1_2_2_1_5"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</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>

<a NAME="E2:45_2_2_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">CR</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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="E3:45_2_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A50</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E4:45_2_2_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E42">A51</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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="c3">CR</font></span>)</span>

;<br/>
<span class="kw">assume </span><a NAME="E6:45_2_2_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> not <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1</span><br/>

<a NAME="E7:45_2_2_1_1_2_2"/><span class="kw">then </span>
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</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="#E5:45"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E6:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="txt" href="#E5:45_2_2_1_1_2_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E3:45_2_2_1_1_2_2"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E8:45_2_2_1_1_2_2"/><span class="kw">then </span>
not <font color="Maroon" title="c3">CR</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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>,1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:45_2_2_1_1_2_2"/>
not <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#R2" title="FINSEQ_8:pred.2">is_substring_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="filerec1.html#T42" target="_self" title="FILEREC1:th.42">Th42</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><a NAME="E17:45_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A52</font></span>: 
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">CR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D10" title="FINSEQ_8:def.10">FINSEQ_8:def 10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_2_3"><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="E1:45_2_2_1_1_2_3"/>
( 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"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> or 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E42">A53</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span><br/><div class="add"><a NAME="E2:45_2_2_1_1_2_3_1"/>
( <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/><a NAME="E3:45_2_2_1_1_2_3_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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="c2">f</font>,<font color="Maroon" title="c3">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="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E42">A53</font></span></a>, <a class="ref" href="finseq_6.html#T122" title="FINSEQ_6:th.122">FINSEQ_6:122</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:45_2_2_1_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A54</font></span>: 
( <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>,1 or  <a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">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="E5:45_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E17:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="ref" href="finseq_8.html#D12" title="FINSEQ_8:def.12">FINSEQ_8:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:45_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E43">A54</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="45_2_2_1_1_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_2_3_2"/>
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span><br/><div class="add"><a NAME="E2:45_2_2_1_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A55</font></span>: 
<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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <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="E3:45_2_2_1_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A56</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:45_2_2_1_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A57</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="finseq_8.html#R5" title="FINSEQ_8:pred.5">is_terminated_by</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_8.html#D12" title="FINSEQ_8:def.12">FINSEQ_8:def 12</a></span>;<br/><a NAME="E5:45_2_2_1_1_2_3_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_2_1_1_2_3_2"><span class="lab"><font color="Green" title="E42">A55</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E6:45_2_2_1_1_2_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E7:45_2_2_1_1_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><a NAME="E8:45_2_2_1_1_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#K1" title="FINSEQ_8:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_2_1_1_2_3_2"><span class="lab"><font color="Green" title="E43">A56</font></span></a>, <a class="ref" href="finseq_8.html#D7" title="FINSEQ_8:def.7">FINSEQ_8:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:45_2_2_1_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K9" title="FINSEQ_8:func.9">addcr</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font></span>)</span>,<font color="Maroon" title="c3">CR</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2_2_1_1_2_3_2"><span class="lab"><font color="Green" title="E44">A57</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:45_2_2"/>
 ex <font color="Olive" title="b1">r</font> being   <a href="filerec1.html#NM1" title="FILEREC1:NM.1">File</a> of <font color="Maroon" title="c1">D</font> st <font color="Olive" title="b1">r</font> <a href="filerec1.html#R1" title="FILEREC1:pred.1">is_a_record_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>
 ; <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>
