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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st (  for <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">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="b2">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</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="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; (  for <font color="Olive" title="b2">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="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">a</font> ) holds <br/> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c1">fp</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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">gcd</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 ) &amp; (  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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><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> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> ) implies  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c1">fp</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> )</span><br/>



<span class="kw">defpred </span><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> ] <span class="kw">means </span>( (  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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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">gcd</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 ) &amp; (  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#K1" title="RELSET_1:func.1">dom</a> $1 holds <br/>$1 <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> ) );<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span>( <font color="Maroon">S<sub>1</sub></font>[$1] implies <font color="Maroon">S<sub>2</sub></font>[$1] );<br/>
<div><span class="lab"><font color="Green" title="E35">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d</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="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>] )</span><br/><span class="kw">assume </span><a NAME="E1:52_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">set </span><font color="Maroon" title="c5">fp1</font> = <font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><a NAME="E2:52_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">fp</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:52_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><a NAME="E2:52_1_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">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">gcd</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E1:52_1_1_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:52_1_1_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E3:52_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">fp</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:52_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">fp</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T39" title="XREAL_1:th.39">XREAL_1:39</a></span>;<br/>
<a NAME="E5:52_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/>
<a NAME="E6:52_1_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="finseq_2.html#T15" title="FINSEQ_2:th.15">FINSEQ_2:15</a></span>;<br/>
<a NAME="E7:52_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E5:52_1_1_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E1:52_1_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/>
<a NAME="E8:52_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">fp</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/>
<a NAME="E9:52_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:52_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52_1_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:52_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c3">fp</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T36" title="WSIERP_1:th.36">WSIERP_1:36</a></span>;<br/><a NAME="E4:52_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c3">fp</font>,<font color="Maroon" title="c4">d</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/><a NAME="E5:52_1_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></span>;<br/><a NAME="E6:52_1_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">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="c3">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">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52_1_1_2"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> )
 <span class="kw">and </span><br/><a NAME="E2:52_1_1_2"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E3:52_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:52_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E5:52_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E1:52_1_1_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="txt" href="#E2:52_1_1_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:52_1_1"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> holds <br/><font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font> implies <font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:52_1_1_3"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E2:52_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:52_1_1_3"/>
<font color="Maroon" title="c3">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a>, <a class="txt" href="#E5:52_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E1:52_1_1_3"><span class="lab"><font color="Green" title="E43">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:52_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="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><a NAME="E9:52_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E38">A4</font></span></a></span>;<br/><a NAME="E10:52_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&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="c3">fp</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>;<br/><a NAME="E11:52_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></span>;<br/><a NAME="E12:52_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c3">fp</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E6:52_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E7:52_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E4:52_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="pepin.html#T4" title="PEPIN:th.4">PEPIN:4</a></span>;<br/><span class="kw">hence </span><a NAME="E13:52_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T96" title="RVSUM_1:th.96">RVSUM_1:96</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:52_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c3">fp</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:52"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<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>]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a>, <a class="ref" href="rvsum_1.html#T94" title="RVSUM_1:th.94">RVSUM_1:94</a></span>;<br/>
<a NAME="E3:52"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_2.html#S2" title="FINSEQ_2:sch.2">FINSEQ_2:sch 2</a>(<a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E35">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:52"/>
( (  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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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">gcd</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 ) &amp; (  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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> holds <br/><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> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> ) implies  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c1">fp</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
