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

<b>let </b><font color="Maroon" title="c1">fp</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">d</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 ;<br/>

<a NAME="E2:41"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="finseq_3.html#T118" title="FINSEQ_3:th.118">FINSEQ_3:118</a></i>;<br/>
<a NAME="E3:41"/><b>then </b><i><font color="Green" title="E31">A3</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"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E4:41"/><i><font color="Green" title="E32">A4</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="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E5:41"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="finseq_3.html#T24" title="FINSEQ_3:th.24">FINSEQ_3:24</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green" title="E34">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E3:41"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="int_4.html#E4:41"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="finseq_3.html#T118" title="FINSEQ_3:th.118">FINSEQ_3:118</a></i>;<br/>
<a NAME="E7:41"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E35">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> )
 ;<br/>

<a NAME="E2:41_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="int_4.html#E3:41"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="int_4.html#E6:41"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="int_4.html#E1:41_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:41_1"/><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/><i><font color="Green" title="E38">A10</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:41_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E1:41_1_1_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:41_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</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="E5:41_1_1_1"/><b>then </b><i><font color="Green" title="E39">A11</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E2:41"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="int_4.html#E1:41_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">w</font> = <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon" title="c7">fpd</font> = <font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>;<br/><a NAME="E7:41_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><b>then </b><i><font color="Green" title="E40">A12</font></i>: <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="int_4.html#E3:41_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="int_4.html#E1:41_1_1_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></i>
<br/>.= 
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="int_4.html#E2:41_1"><i><font color="Green" title="E36">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"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="int_4.html#E5:41_1_1_1"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="int_4.html#E3:41_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="int_4.html#E1:41_1_1_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></i>
;<br/><b>hence </b><a NAME="E10:41_1_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="int_4.html#E8:41_1_1_1"><i><font color="Green" title="E40">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/><i><font color="Green" title="E38">A13</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_2"/><b>then </b><i><font color="Green" title="E39">A14</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E3:41"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="int_4.html#E4:41"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="int_4.html#E5:41"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="int_4.html#E6:41"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="int_4.html#E1:41_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="int_4.html#E3:41_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_2_1"/>
( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="int_4.html#E3:41"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="int_4.html#E6:41"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="int_4.html#E1:41_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_2_1_1"/><i><font color="Green" title="E40">A15</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp; <font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:41_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E41">A16</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">fp</font>
 <b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:41_1_1_2_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_4.html#E2:41"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="int_4.html#E1:41_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="int_4.html#E1:41_1_1_2_1_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="int_4.html#E3:41_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="int_4.html#E1:41_1_1_2"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="int_4.html#E1:41_1_1_2_1_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>
;<br/><b>hence </b><a NAME="E6:41_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="int_4.html#E2:41_1_1_2"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="int_4.html#E3:41_1_1_2_1_1"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2_1_2"/><i><font color="Green" title="E40">A17</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 ;<br/><div class="add"><a NAME="E2:41_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">d</font>
 <b>by </b><i><a class="txt" href="int_4.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="int_4.html#E2:41_1_1_2"><i><font color="Green" title="E39">A14</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="int_4.html#E2:41"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="int_4.html#E1:41_1_1_2_1_2"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></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>hence </b><a NAME="E8:41"/>
 <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c3">b</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="int_4.html#E6:41"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div>
