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

<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/>





<i><font color="Green" title="E12">A1</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D1" target="_self" title="NAT_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<i><font color="Green" title="E13">A2</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D1" target="_self" title="NAT_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T12" target="_self" title="NAT_3:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E3:21"/><i><font color="Green" title="E14">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D1" target="_self" title="NAT_3:def.1">Def1</a></i>;<br/>
<a NAME="E4:21"/><i><font color="Green" title="E15">X</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:21_3"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:21_3"/><i><font color="Green" title="E17">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 <b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="#E4:21"><i><font color="Green" title="E15">X</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:21_3"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:21_3"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_3_1"/>
( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="#E2:21_3"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_1_1"><b>suppose </b></a><a NAME="E1:21_3_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:21_3_1_1"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:21_3_1_1"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="#E2:21_3"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:21_3_1_1"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="#E2:21_3"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="#E2:21_3_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="#E3:21_3"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="nat_3.html#D1" target="_self" title="NAT_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="#E3:21_3_1_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E4:21_3_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="nat_3.html#D1" target="_self" title="NAT_3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E4:21_3_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_3_1_2"><b>suppose </b></a><a NAME="E1:21_3_1_2"/><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="#E3:21_3"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="nat_3.html#D1" target="_self" title="NAT_3:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="#E1:21_3_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="txt" href="#E3:21"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="#E1:21_3_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<b>by </b><i><a class="ref" href="nat_3.html#T12" target="_self" title="NAT_3:th.12">Th12</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="nat_3.html#K2" title="NAT_3:func.2">|^</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E2:21"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
