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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>, <font color="Maroon">CR</font> be    <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font>;<br/>



<b>set </b><font color="Maroon">i0</font> =  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font>;<br/>
<a NAME="E1:44"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/>
 <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
not <font color="Maroon">CR</font> <a href="finseq_8.html#R2">is_substring_of</a>  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>,1
 
<b>by </b><i><a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a></i>;<br/>
<b>hence </b><a NAME="E3:44_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:44"/><b>then </b>
 <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E3:44"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E5:44"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_1"><b>suppose </b></a><a NAME="E1:44_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:44_2_1"/>
<font color="Maroon">CR</font> <a href="filerec1.html#R1" target="_self">is_a_record_of</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>,<font color="Maroon">CR</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:44_2_1"/>
 ex <font color="Olive">r2</font> being   <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font> st <font color="Olive">r2</font> <a href="filerec1.html#R1" target="_self">is_a_record_of</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>
 <b>by </b><i><a class="ref" href="filerec1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2"><b>suppose </b></a><a NAME="E1:44_2_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_2"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> &amp; <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> ) )
 <b>by </b><i><a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a>, <a class="ref" href="filerec1.html#T2" target="_self">Th2</a></i>;<br/><b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E3:44_2_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#R5">is_terminated_by</a> <font color="Maroon">CR</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_8.html#D12">FINSEQ_8:def 12</a></i>;<br/><a NAME="E4:44_2_2_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E3:44"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/><a NAME="E5:44_2_2_1_1_1"/><b>then </b>
<font color="Maroon">CR</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K3">ovlpart</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font>
 <b>by </b><i><a class="ref" href="finseq_6.html#T86">FINSEQ_6:86</a></i>;<br/><a NAME="E6:44_2_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_8.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">CR</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="finseq_8.html#K3">ovlpart</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E7:44_2_2_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="finseq_8.html#K4">ovlcon</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#D3">FINSEQ_8:def 3</a></i>;<br/><a NAME="E8:44_2_2_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#D11">FINSEQ_8:def 11</a></i>;<br/><a NAME="E9:44_2_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="filerec1.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E10:44_2_2_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:44_2_2_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies ( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_2_2_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:44_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_2_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T80">FINSEQ_5:80</a>, <a class="ref" href="filerec1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E1:44_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_8.html#D11">FINSEQ_8:def 11</a></i>
<br/>.= 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E1:44_2_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/><b>hence </b><a NAME="E3:44_2_2_1_1_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="filerec1.html#E3:44_2_2"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_1_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><div class="add"><i><font color="Green">E62</font></i>:  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E1:44_2_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="filerec1.html#E1:44_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_8.html#D11">FINSEQ_8:def 11</a></i>
;<br/><a NAME="E3:44_2_2_1_1_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="finseq_5.html#T19">FINSEQ_5:19</a></i>;<br/><b>hence </b><a NAME="E4:44_2_2_1_1_1_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">f</font>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="filerec1.html#E3:44_2_2"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="filerec1.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E12:44_2_2_1_1_1"/><b>then </b>
 <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i><a class="txt" href="filerec1.html#E2:44_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a></i>;<br/><b>hence </b><a NAME="E13:44_2_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="filerec1.html#R1" target="_self">is_a_record_of</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_2"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font> )
 <b>by </b><i>, <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a></i>;<br/><a NAME="E3:44_2_2_1_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E4:44_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E3:44"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>;<br/><a NAME="E5:44_2_2_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T18">FINSEQ_5:18</a></i>;<br/><a NAME="E6:44_2_2_1_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T28">FINSEQ_5:28</a></i>;<br/><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_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">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:44_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">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E3:44_2_2_1_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font>
 <b>by </b><i>, <a class="txt" href="filerec1.html#E1:44_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:44_2_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:44_2_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:44_2_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font>
 <b>by </b><i><a class="txt" href="filerec1.html#E3:44_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:44_2_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> &amp; <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 <b>by </b><i><a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a>, <a class="ref" href="filerec1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:44_2_2_1_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_2_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>
 ;<br/><a NAME="E2:44_2_2_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><b>hence </b><a NAME="E3:44_2_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="txt" href="filerec1.html#E3:44_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E11:44_2_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/><a NAME="E12:44_2_2_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="txt" href="filerec1.html#E3:44_2_2"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E13:44_2_2_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E14:44_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E15:44_2_2_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="filerec1.html#T3" target="_self">Th3</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E16:44_2_2_1_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E17:44_2_2_1_1_2"/>
( ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</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">len</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> ) ) ) &amp; ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> 0 implies  not <font color="Maroon">CR</font> <a href="finseq_8.html#R2">is_substring_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:44_2_2_1_1_2_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</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">len</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E3:44_2_2_1_1_2_3_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:44_2_2_1_1_2_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 ;<br/>

<a NAME="E5:44_2_2_1_1_2_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green">E102</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>

;<br/>
<a NAME="E8:44_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E9:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</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">len</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="filerec1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E11:44_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E12:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E13:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green">E104</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="filerec1.html#T6" target="_self">Th6</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="ref" href="filerec1.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font>

;<br/>
<a NAME="E15:44_2_2_1_1_2_3_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="ref" href="filerec1.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan3.html#T21">JORDAN3:21</a></i>;<br/>
<b>set </b><font color="Maroon">f2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/>
<a NAME="E16:44_2_2_1_1_2_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E17:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E18:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E19:44_2_2_1_1_2_3_1"/>
 <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E20:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="filerec1.html#T3" target="_self">Th3</a>, <a class="ref" href="filerec1.html#T14" target="_self">Th14</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E21:44_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E22:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<i><font color="Green">E111</font></i>:  <a href="finseq_1.html#K3">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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>
<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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="filerec1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>
<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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="ref" href="filerec1.html#T18" target="_self">Th18</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font>

;<br/>
<a NAME="E24:44_2_2_1_1_2_3_1"/>
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font>
 
<b>by </b><i><a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a></i>;<br/>
<a NAME="E25:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a>, <a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E26:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/>
<a NAME="E27:44_2_2_1_1_2_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="jordan3.html#K1">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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font> )
 
<b>by </b><i><a class="ref" href="jordan3.html#T25">JORDAN3:25</a>, <a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E28:44_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</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">len</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i>, <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a></i>;<br/>

<a NAME="E29:44_2_2_1_1_2_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E30:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E31:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E32:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E33:44_2_2_1_1_2_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E34:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green">E113</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>

;<br/>
<a NAME="E36:44_2_2_1_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E37:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E38:44_2_2_1_1_2_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E39:44_2_2_1_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E40:44_2_2_1_1_2_3_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>thus </b><a NAME="E41:44_2_2_1_1_2_3_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Olive">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1_1_2_3_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:44_2_2_1_1_2_3_1_5"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  )
 ;<br/>

<a NAME="E2:44_2_2_1_1_2_3_1_5"/><i><font color="Green">E117</font></i>: 
 <a href="jordan3.html#K1">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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T25">JORDAN3:25</a>, <a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a></i>;<br/>
<i><font color="Green">E118</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T21">JORDAN3:21</a></i>
;<br/>
<a NAME="E4:44_2_2_1_1_2_3_1_5"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2_1_1_2_3_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2_2_1_1_2_3_1_5_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 ;<br/><a NAME="E2:44_2_2_1_1_2_3_1_5_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 <b>by </b><i><a class="ref" href="filerec1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:44_2_2_1_1_2_3_1_5_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font>
 <b>by </b><i>, <a class="ref" href="filerec1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, , </i>;<br/><a NAME="E4:44_2_2_1_1_2_3_1_5_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="filerec1.html#E1:44_2_2_1_1_2"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="sprect_3.html#T9">SPRECT_3:9</a></i>;<br/><a NAME="E5:44_2_2_1_1_2_3_1_5_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E6:44_2_2_1_1_2_3_1_5_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><b>then </b><i><font color="Green">E125</font></i>: <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E8:44_2_2_1_1_2_3_1_5_2"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:44_2_2_1_1_2_3_1_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:44_2_2_1_1_2_3_1_5_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a>, <a class="txt" href="filerec1.html#E3:44_2_2_1_1_2"><i><font color="Green">E65</font></i></a>, </i>;<br/><i><font color="Green">E127</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T21">JORDAN3:21</a></i>
<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">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="filerec1.html#T19" target="_self">Th19</a>, , <a class="ref" href="finseq_5.html#T80">FINSEQ_5:80</a></i>
<br/>.= 
<font color="Maroon">CR</font>
<b>by </b><i><a class="ref" href="filerec1.html#T19" target="_self">Th19</a>, , <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a>, <a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a>, , </i>
;<br/><a NAME="E12:44_2_2_1_1_2_3_1_5_2"/><i><font color="Green">E128</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a>, <a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E13:44_2_2_1_1_2_3_1_5_2"/><i><font color="Green">E129</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="filerec1.html#E2:44_2_2_1_1_2"><i><font color="Green">E64</font></i></a>, </i>;<br/><a NAME="E14:44_2_2_1_1_2_3_1_5_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font> )
 <b>by </b><i><a class="txt" href="filerec1.html#E5:44_2_2_1_1_2"><i><font color="Green">E70</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, , <a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>;<br/><a NAME="E15:44_2_2_1_1_2_3_1_5_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a> 
 <b>by </b><i><a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a></i>;<br/><b>hence </b><a NAME="E16:44_2_2_1_1_2_3_1_5_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:44_2_2_1_1_2_3_1_5"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:44_2_2_1_1_2_3"/><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:44_2_2_1_1_2_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i><a class="txt" href="filerec1.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E4:44_2_2_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>then </b><i><font color="Green">E131</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>

;<br/>
<a NAME="E6:44_2_2_1_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E7:44_2_2_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E8:44_2_2_1_1_2_3"/><b>then </b>
 <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="filerec1.html#T6" target="_self">Th6</a>, <a class="ref" href="filerec1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a>, </i>;<br/>
<a NAME="E9:44_2_2_1_1_2_3"/><b>then </b><i><font color="Green">E132</font></i>: 
not <font color="Maroon">CR</font> <a href="finseq_8.html#R2">is_substring_of</a>  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>,1
 
<b>by </b><i><a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a></i>;<br/>
<a NAME="E10:44_2_2_1_1_2_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E11:44_2_2_1_1_2_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E12:44_2_2_1_1_2_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:44_2_2_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E14:44_2_2_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K7">instr</a> 1,<span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font>
 
<b>by </b><i><a class="ref" href="filerec1.html#T10" target="_self">Th10</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:44_2_2_1_1_2_3"/>
not <font color="Maroon">CR</font> <a href="finseq_8.html#R2">is_substring_of</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1
 <b>by </b><i>, <a class="txt" href="filerec1.html#E1:44"><i><font color="Green">E33</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E18:44_2_2_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> &amp;  <a href="finseq_8.html#K7">instr</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">CR</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span> )
 <b>by </b><i><a class="txt" href="filerec1.html#E2:44_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_8.html#D10">FINSEQ_8:def 10</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_2_1_1_2_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> or 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2_4_1"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_2_4_1"/><i><font color="Green">E135</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1_2_4_1"/><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="filerec1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E3:44_2_2_1_1_2_4_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T18">FINSEQ_5:18</a></i>;<br/><a NAME="E4:44_2_2_1_1_2_4_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies ( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> &amp;  <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, , <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>;<br/><a NAME="E5:44_2_2_1_1_2_4_1"/><b>then </b>
( ( <font color="Maroon">CR</font> <a href="finseq_8.html#K1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_8.html#R2">is_substring_of</a>  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>,1 or  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font> <a href="finseq_8.html#NR3" target="_self">is_preposition_of</a>  ) &amp; <span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#R5">is_terminated_by</a> <font color="Maroon">CR</font> )
 <b>by </b><i><a class="ref" href="filerec1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_8.html#D12">FINSEQ_8:def 12</a>, <a class="ref" href="finseq_8.html#D8">FINSEQ_8:def 8</a></i>;<br/><b>hence </b><a NAME="E6:44_2_2_1_1_2_4_1"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="filerec1.html#R1" target="_self">is_a_record_of</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1_1_2_4_2"><b>suppose </b></a><a NAME="E1:44_2_2_1_1_2_4_2"/>
1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:44_2_2_1_1_2_4_2"/><b>then </b><i><font color="Green">E137</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p0">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:44_2_2_1_1_2_4_2"/><b>then </b><i><font color="Green">E138</font></i>: 
0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:44_2_2_1_1_2_4_2"/><i><font color="Green">E139</font></i>: 
0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E5:44_2_2_1_1_2_4_2"/><i><font color="Green">E140</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="txt" href="filerec1.html#E2:44_2_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E6:44_2_2_1_1_2_4_2"/><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="finseq_8.html#R5">is_terminated_by</a> <font color="Maroon">CR</font>
 <b>by </b><i><a class="ref" href="finseq_8.html#D12">FINSEQ_8:def 12</a>, </i>;<br/><a NAME="E7:44_2_2_1_1_2_4_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="filerec1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E8:44_2_2_1_1_2_4_2"/><b>then </b>
<font color="Maroon">CR</font> <a href="finseq_8.html#K1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CR</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E9:44_2_2_1_1_2_4_2"/><b>then </b>
<font color="Maroon">CR</font> <a href="finseq_8.html#K1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p5">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_8.html#R2">is_substring_of</a>  <a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>,1
 <b>by </b><i>, <a class="ref" href="finseq_8.html#D7">FINSEQ_8:def 7</a></i>;<br/><b>hence </b><a NAME="E10:44_2_2_1_1_2_4_2"/>
<span class="p1">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span> <a href="finseq_1.html#K17">|</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">instr</a> 1,<span class="p4">(<span class="default"><a href="finseq_8.html#K8">addcr</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font></span>)</span>,<font color="Maroon">CR</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">CR</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="filerec1.html#R1" target="_self">is_a_record_of</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>thus </b><a NAME="E4:44_2_2"/>
 ex <font color="Olive">r</font> being   <a href="filerec1.html#NM1" target="_self">File</a> of <font color="Maroon">D</font> st <font color="Olive">r</font> <a href="filerec1.html#R1" target="_self">is_a_record_of</a> <font color="Maroon">f</font>,<font color="Maroon">CR</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
