<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b>for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <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>   st <font color="Olive" title="b1">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> $1 &amp; <font color="Olive" title="b2">c</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> $1 &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> holds <br/><span class="p1">(<span class="default">$1 <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default">$1 <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ] <b>means </b>for <font color="Olive" title="b1">b</font> being   <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>   st <font color="Olive" title="b1">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> $1 holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> $1,<font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default">$1 <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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> ] <b>means </b>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon">S<sub>1</sub></font>[$1] implies <font color="Maroon">S<sub>2</sub></font>[$1] );<br/>
<a NAME="E1:42"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">&lt;*&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ]
 
;<br/>
<div><i><font color="Green" title="E31">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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="c3">d</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="E1:42_1"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">fp</font>]
 ;<br/><b>set </b><font color="Maroon" title="c4">fp1</font> = <font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>;<br/><b>set </b><font color="Maroon" title="c5">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green" title="E33">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="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>] )
 ;<br/><a NAME="E2:42_1_1"/><i><font color="Green" title="E34">A5</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="c2">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="c3">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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_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="E1:42_1_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="c2">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="c3">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> 2 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1"/><i><font color="Green" title="E35">A6</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="c2">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="c3">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> 2
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">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> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">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> 2</span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:42_1_1_1_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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> 2</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_1_1_1_1_1"/>
( 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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> &amp; 2 <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="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> &amp; 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 )
 
<b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E2:42_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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> 2</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1"><i><font color="Green" title="E33">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:42_1_1_1_1_1"/>
for <font color="Olive" title="b1">b</font> being   <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>   st <font color="Olive" title="b1">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="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</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="E1:42_1_1_1_1_1_2"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c6">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="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E2:42_1_1_1_1_1_2"/><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1_1_1_1_2"><i><font color="Green" title="E38">A9</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="hsNdiv(this)" href="javascript:()" title="42_1_1_1_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="E1:42_1_1_1_1_1_2_1"/>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="int_4.html#E2:42_1_1_1_1_1_2"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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="c6">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">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> 2</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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> 1</span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E2:42_1_1_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="wsierp_1.html#T26" title="WSIERP_1:th.26">WSIERP_1:26</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="int_4.html#E3:42_1_1_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="rvsum_1.html#T125" title="RVSUM_1:th.125">RVSUM_1:125</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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="c6">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p5"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">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> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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> 2</span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E2:42_1_1_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="wsierp_1.html#T26" title="WSIERP_1:th.26">WSIERP_1:26</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="int_4.html#E3:42_1_1_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="rvsum_1.html#T125" title="RVSUM_1:th.125">RVSUM_1:125</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E5:42_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E35">A11</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <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="E3:42_1_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:42_1_1_1_1_2"/><i><font color="Green" title="E36">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font> 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/><a NAME="E7:42_1_1_1_1_2"/><i><font color="Green" title="E37">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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#E5:42_1_1_1_1_2"><i><font color="Green" title="E36">A12</font></i></a></i>;<br/><a NAME="E8:42_1_1_1_1_2"/><i><font color="Green" title="E38">A14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">fp</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</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="E1:42_1_1_1_1_2_1"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c8">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="c2">fp</font> &amp; <font color="Maroon" title="c9">c</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="c2">fp</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> )
 ;<br/>

<a NAME="E2:42_1_1_1_1_2_1"/><i><font color="Green" title="E40">A16</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">fp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E3:42_1_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">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="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">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="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">c</font> )
 
<b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1_1_1_2_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="int_4.html#E1:42_1_1_1_1_2_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="txt" href="int_4.html#E2:42_1_1_1_1_2_1"><i><font color="Green" title="E40">A16</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:42_1_1_1_1_2"/><i><font color="Green" title="E39">A17</font></i>: 
for <font color="Olive" title="b1">a</font> being   <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>   st <font color="Olive" title="b1">a</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="c2">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</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="E1:42_1_1_1_1_2_2"/><i><font color="Green" title="E40">A18</font></i>: 
<font color="Maroon" title="c8">a</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="c2">fp</font>
 ;<br/>

<a NAME="E2:42_1_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E41">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">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="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">a</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/>
<a NAME="E3:42_1_1_1_1_2_2"/><i><font color="Green" title="E42">A20</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">fp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E4:42_1_1_1_1_2_2"/><i><font color="Green" title="E43">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E5:42_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</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="E6:42_1_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E44">A22</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1_1_1_2_2"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:42_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">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#E2:42_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><a NAME="E8:42_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E1:42_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="int_4.html#E1:42_1_1_1_1_2_2"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="int_4.html#E2:42_1_1_1_1_2_2"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="int_4.html#E3:42_1_1_1_1_2_2"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="int_4.html#E4:42_1_1_1_1_2_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="int_4.html#E6:42_1_1_1_1_2_2"><i><font color="Green" title="E44">A22</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:42_1_1_1_1_2"/>
for <font color="Olive" title="b1">b</font> being   <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>   st <font color="Olive" title="b1">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="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">b</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="E1:42_1_1_1_1_2_3"/><i><font color="Green" title="E40">A23</font></i>: 
<font color="Maroon" title="c8">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="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E2:42_1_1_1_1_2_3"/><i><font color="Green" title="E41">A24</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">b</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="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 
<b>by </b><i><a class="txt" href="int_4.html#E2:42_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="int_4.html#E1:42_1_1_1_1_2_3"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1_1_2_3_1"/>
( <font color="Maroon" title="c8">b</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="c2">fp</font> or <font color="Maroon" title="c8">b</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="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="int_4.html#E2:42_1_1_1_1_2_3"><i><font color="Green" title="E41">A24</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="42_1_1_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c8">b</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="c2">fp</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E42">A25</font></i>: 
<font color="Maroon" title="c8">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="c2">fp</font>
 <b>by </b><i><a class="txt" href="int_4.html#E2:42_1_1_1_1_2_3"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:42_1_1_1_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E43">A26</font></i>: 
 <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">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="c2">fp</font>,<font color="Maroon" title="c8">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="c3">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#E7:42_1_1_1_1_2"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="int_4.html#T30" target="_self" title="INT_4:th.30">Th30</a></i>;<br/><a NAME="E4:42_1_1_1_1_2_3_1_1"/><i><font color="Green" title="E44">A27</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c2">fp</font>,<font color="Maroon" title="c8">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E1:42_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="int_4.html#E2:42_1_1_1_1_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="int_4.html#E8:42_1_1_1_1_2"><i><font color="Green" title="E38">A14</font></i></a>, <a class="txt" href="int_4.html#E2:42_1_1_1_1_2_3_1_1"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:42_1_1_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E9:42_1_1_1_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="int_4.html#E2:42_1_1_1_1_2_3_1_1"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/><a NAME="E6:42_1_1_1_1_2_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c2">fp</font>,<font color="Maroon" title="c8">b</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E4:42_1_1_1_1_2_3_1_1"><i><font color="Green" title="E44">A27</font></i></a>, <a class="ref" href="wsierp_1.html#T12" title="WSIERP_1:th.12">WSIERP_1:12</a></i>;<br/><a NAME="E7:42_1_1_1_1_2_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <font color="Maroon" title="c2">fp</font>,<font color="Maroon" title="c8">b</font></span>)</span> <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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="rvsum_1.html#T126" title="RVSUM_1:th.126">RVSUM_1:126</a></i>;<br/><b>hence </b><a NAME="E8:42_1_1_1_1_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="int_4.html#E2:42_1_1_1_1_2_3_1_1"><i><font color="Green" title="E42">A25</font></i></a>, <a class="txt" href="int_4.html#E3:42_1_1_1_1_2_3_1_1"><i><font color="Green" title="E43">A26</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="42_1_1_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c8">b</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="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c8">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c3">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="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><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"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T48" title="WSIERP_1:th.48">WSIERP_1:48</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="int_4.html#E9:42_1_1_1_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="wsierp_1.html#T43" title="WSIERP_1:th.43">WSIERP_1:43</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E11:42_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:42_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:42_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">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="c3">d</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:42"/>
for <font color="Olive" title="b1">fp</font> being   <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>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">fp</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2" title="FINSEQ_2:sch.2">FINSEQ_2:sch 2</a>(<a class="txt" href="int_4.html#E1:42"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="int_4.html#E2:42"><i><font color="Green" title="E31">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:42"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; ( for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</font> being   <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>   st <font color="Olive" title="b1">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; <font color="Olive" title="b2">c</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; <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) implies for <font color="Olive" title="b1">b</font> being   <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>   st <font color="Olive" title="b1">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> holds <br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <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="Olive" title="b1">b</font></span>)</span></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>


</div>
