<?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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="E1:43"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1"/>
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:43_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#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('finseq_8/D10')" onmouseout="rh()">FINSEQ_8:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filerec1.html#T40" target="_self" onmouseover="rs('filerec1/T40')" onmouseout="rh()">Th40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:43"/><span class="kw">then </span>
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E3:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E4:43"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E5:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_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#K5" title="NUMBERS:func.5">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1"><span class="kw">suppose </span></a><a NAME="E1:43_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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="E2:43_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="E3:43_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#T22" target="_self" onmouseover="rs('filerec1/T22')" onmouseout="rh()">Th22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_2"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E1:43"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="finseq_8.html#D10" onmouseover="rs('finseq_8/D10')" onmouseout="rh()">FINSEQ_8:def 10</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_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#K5" title="NUMBERS:func.5">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_2_1_1_1"/><span class="lab"><font color="Green" title="E31">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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">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>
 ;<br/><a NAME="E3:43_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#K3" title="FINSEQ_8:func.3">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#T80" onmouseover="rs('finseq_6/T80')" onmouseout="rh()">FINSEQ_6:80</a></span>;<br/><a NAME="E4:43_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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#K3" title="FINSEQ_8:func.3">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="#E2:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/><a NAME="E5:43_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="finseq_8.html#K4" title="FINSEQ_8:func.4">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" onmouseover="rs('finseq_8/D3')" onmouseout="rh()">FINSEQ_8:def 3</a></span>;<br/><a NAME="E6:43_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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>;<br/><a NAME="E7:43_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:43_2_2_1_1_1"/><span class="lab"><font color="Green" title="E35">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a>  implies ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="43_2_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E1:43_2_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="finseq_5.html#T77" onmouseover="rs('finseq_5/T77')" onmouseout="rh()">FINSEQ_5:77</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E5:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="finseq_8.html#D11" onmouseover="rs('finseq_8/D11')" onmouseout="rh()">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E2:43_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E7:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#T17" onmouseover="rs('finseq_5/T17')" onmouseout="rh()">FINSEQ_5:17</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E2:43_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E5:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="finseq_8.html#D11" onmouseover="rs('finseq_8/D11')" onmouseout="rh()">FINSEQ_8:def 11</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E7:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E1:43_2_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E2:43_2_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E9:43_2_2_1_1_1"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="finseq_8.html#D12" onmouseover="rs('finseq_8/D12')" onmouseout="rh()">FINSEQ_8:def 12</a></span>;<br/><a NAME="E10:43_2_2_1_1_1"/>
 <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E5:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="finseq_8.html#D11" onmouseover="rs('finseq_8/D11')" onmouseout="rh()">FINSEQ_8:def 11</a></span>;<br/><a NAME="E11:43_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="txt" href="#E8:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E12:43_2_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E9:43_2_2_1_1_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_2_1_1_2"/><span class="lab"><font color="Green" title="E31">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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_2"/>
<font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E1:43"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="finseq_8.html#D10" onmouseover="rs('finseq_8/D10')" onmouseout="rh()">FINSEQ_8:def 10</a></span>;<br/><a NAME="E3:43_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#T43" onmouseover="rs('nat_1/T43')" onmouseout="rh()">NAT_1:43</a></span>;<br/><a NAME="E4:43_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="card_1.html#T27" onmouseover="rs('card_1/T27')" onmouseout="rh()">CARD_1:27</a>, <a class="ref" href="rfinseq.html#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>;<br/><a NAME="E5:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</a></span>;<br/><a NAME="E6:43_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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="#E3:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E7:43_2_2_1_1_2"/><span class="lab"><font color="Green" title="E34">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#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E2:43_2_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><a NAME="E9:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>;<br/><a NAME="E10:43_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#T16" onmouseover="rs('finseq_5/T16')" onmouseout="rh()">FINSEQ_5:16</a></span>;<br/><a NAME="E11:43_2_2_1_1_2"/><span class="lab"><font color="Green" title="E36">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#T25" onmouseover="rs('finseq_5/T25')" onmouseout="rh()">FINSEQ_5:25</a></span>;<br/><div><span class="lab"><font color="Green" title="E37">A22</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_2_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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="43_2_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">addcr</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E10:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A20</font></span></a>, <a class="txt" href="#E11:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E36">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E38">A23</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">addcr</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E3:43_2_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E4:43_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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E1:43_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A23</font></span></a>, <a class="txt" href="#E3:43_2_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A24</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E13:43_2_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><a NAME="E14:43_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">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#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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="#E5:43"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><a NAME="E15:43_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E16:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> &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#NK1" title="NUMBERS:NK.1">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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> &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#NK1" title="NUMBERS:NK.1">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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2_2_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> &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#NK1" title="NUMBERS:NK.1">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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_2_2_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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> &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#NK1" title="NUMBERS:NK.1">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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="E4:43_2_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E40">A27</font></span>: 
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<a NAME="E5:43_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E6:43_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A28</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E7:43_2_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E42">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E8:43_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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E9:43_2_2_1_1_2_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E5:43"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E15:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E4:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E43">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#D1" onmouseover="rs('rfinseq/D1')" onmouseout="rh()">RFINSEQ:def 1</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E14:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E7:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E6:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">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="E11:43_2_2_1_1_2_2_1"/><span class="lab"><font color="Green" title="E44">A31</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E12:43_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a></span>;<br/>
<a NAME="E13:43_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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T42" onmouseover="rs('nat_d/T42')" onmouseout="rh()">NAT_D:42</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E5:43"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E11:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">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="E15:43_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A32</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">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#T80" onmouseover="rs('finseq_5/T80')" onmouseout="rh()">FINSEQ_5:80</a></span>;<br/>
<a NAME="E16:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E2:43_2_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/>
<a NAME="E17:43_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">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="txt" href="#E15:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>;<br/>
<a NAME="E18:43_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">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>;<br/>
<a NAME="E19:43_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>;<br/>
<a NAME="E20:43_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E47">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>

;<br/>
<a NAME="E22:43_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E23:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E15:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E24:43_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/>
<a NAME="E25:43_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E26:43_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A36</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E11:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>

;<br/>
<a NAME="E28:43_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">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E5:43"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E26:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:43_2_2_1_1_2_2_1"/>
<font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E10:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E19:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#NK1" title="NUMBERS:NK.1">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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="E30:43_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E31:43_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E32:43_2_2_1_1_2_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E33:43_2_2_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E34:43_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#NK1" title="NUMBERS:NK.1">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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_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#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="E1:43_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E52">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="E2:43_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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:43_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="E54">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_d/T34')" onmouseout="rh()">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#T80" onmouseover="rs('finseq_5/T80')" onmouseout="rh()">FINSEQ_5:80</a></span>
;<br/>
<a NAME="E5:43_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E55">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>;<br/>
<a NAME="E6:43_2_2_1_1_2_2_1_5"/><span class="lab"><font color="Green" title="E56">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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2_2_1_1_2_2_1_5_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_2_2_1_1_2_2_1_5_2"/><span class="lab"><font color="Green" title="E57">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:43_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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E5:43"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E21:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E31:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E3:43_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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E1:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="nat_d.html#T56" onmouseover="rs('nat_d/T56')" onmouseout="rh()">NAT_D:56</a></span>;<br/><a NAME="E4:43_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('filerec1/T4')" onmouseout="rh()">Th4</a></span>;<br/><a NAME="E5:43_2_2_1_1_2_2_1_5_2"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A47</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">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#K1" title="NAT_1:func.1">+</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="#E6:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
;<br/><a NAME="E7:43_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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="#E14:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E1:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E8:43_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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E31:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E9:43_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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E10:43_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A48</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E33:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E5:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#T80" onmouseover="rs('finseq_5/T80')" onmouseout="rh()">FINSEQ_5:80</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">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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="#E6:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="txt" href="#E10:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="ref" href="finseq_5.html#T77" onmouseover="rs('finseq_5/T77')" onmouseout="rh()">FINSEQ_5:77</a></span>
<br/>.= 
<font color="Maroon" title="c3">CR</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E3:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E5:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E4:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E6:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>
;<br/><a NAME="E12:43_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">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#K8" title="FINSEQ_8:func.8">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="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a></span>;<br/><a NAME="E13:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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="#E2:43_2_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><a NAME="E14:43_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#K8" title="FINSEQ_8:func.8">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="#E4:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E15:43_2_2_1_1_2_2_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">CR</font> <a href="finseq_8.html#NR3" title="FINSEQ_8:NR.3">is_preposition_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="finseq_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E16:43_2_2_1_1_2_2_1_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:43_2_2_1_1_2_2_1"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E1:43_2_2_1_1_2_2_1_5"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E1:43_2_2_1_1_2_2_1_5_2"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="finseq_8.html#D10" onmouseover="rs('finseq_8/D10')" onmouseout="rh()">FINSEQ_8:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>

<a NAME="E2:43_2_2_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E3:43_2_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A50</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E4:43_2_2_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="nat_1.html#T12" onmouseover="rs('nat_1/T12')" onmouseout="rh()">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E41">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">CR</font></span>)</span>

;<br/>
<span class="kw">assume </span><a NAME="E6:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E7:43_2_2_1_1_2_2"/><span class="kw">then </span>
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="txt" href="#E5:43_2_2_1_1_2_2"><span class="lab"><font color="Green" title="E41">A51</font></span></a>, <a class="txt" href="#E3:43_2_2_1_1_2_2"><span class="lab"><font color="Green" title="E40">A50</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<a NAME="E8:43_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#K8" title="FINSEQ_8:func.8">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" onmouseover="rs('finseq_8/D10')" onmouseout="rh()">FINSEQ_8:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E1:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="filerec1.html#T41" target="_self" onmouseover="rs('filerec1/T41')" onmouseout="rh()">Th41</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E17:43_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A52</font></span>: 
 <a href="finseq_8.html#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</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" onmouseover="rs('finseq_8/D10')" onmouseout="rh()">FINSEQ_8:def 10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_2_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="43_2_2_1_1_2_3_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_2_1_1_2_3_1"/><span class="lab"><font color="Green" title="E41">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_2_3_1"/>
( <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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="#E14:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E3:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K5" title="NUMBERS:func.5">0</a>  implies ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">addcr</a> (<font color="Maroon" title="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#K8" title="FINSEQ_8:func.8">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E1:43_2_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E41">A53</font></span></a>, <a class="ref" href="finseq_6.html#T116" onmouseover="rs('finseq_6/T116')" onmouseout="rh()">FINSEQ_6:116</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:43_2_2_1_1_2_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A54</font></span>: 
( <font color="Maroon" title="c3">CR</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">addcr</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">CR</font>),1 or <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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>  <a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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_8.html#D8" onmouseover="rs('finseq_8/D8')" onmouseout="rh()">FINSEQ_8:def 8</a></span>;<br/><a NAME="E5:43_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#R4" title="FINSEQ_8:pred.4">is_terminated_by</a> <font color="Maroon" title="c3">CR</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="txt" href="#E17:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E40">A52</font></span></a>, <a class="ref" href="finseq_8.html#D12" onmouseover="rs('finseq_8/D12')" onmouseout="rh()">FINSEQ_8:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:43_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E4:43_2_2_1_1_2_3_1"><span class="lab"><font color="Green" title="E42">A54</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_2_1_1_2_3_2"><span class="kw">suppose </span></a><a NAME="E1:43_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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="E2:43_2_2_1_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A55</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:43_2_2_1_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">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#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_2_2_1_1_2"><span class="lab"><font color="Green" title="E37">A22</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E4:43_2_2_1_1_2_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A57</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#R4" title="FINSEQ_8:pred.4">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" onmouseover="rs('finseq_8/D12')" onmouseout="rh()">FINSEQ_8:def 12</a></span>;<br/><a NAME="E5:43_2_2_1_1_2_3_2"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">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#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E2:43_2_2_1_1_2_3_2"><span class="lab"><font color="Green" title="E41">A55</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E6:43_2_2_1_1_2_3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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> 
 ;<br/><a NAME="E7:43_2_2_1_1_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">CR</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/><a NAME="E8:43_2_2_1_1_2_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">CR</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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#K8" title="FINSEQ_8:func.8">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="#E3:43_2_2_1_1_2_3_2"><span class="lab"><font color="Green" title="E42">A56</font></span></a>, <a class="ref" href="finseq_8.html#D7" onmouseover="rs('finseq_8/D7')" onmouseout="rh()">FINSEQ_8:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_2_2_1_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K7" title="FINSEQ_8:func.7">instr</a> (1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8" title="FINSEQ_8:func.8">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#K1" title="NAT_1:func.1">+</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="#E4:43_2_2_1_1_2_3_2"><span class="lab"><font color="Green" title="E43">A57</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></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="E4:43_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
