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

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a>  holds <br/> <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Olive" title="b1">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c3">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a>  implies  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E2:60"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E3:60"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1"><span class="kw">suppose </span></a><a NAME="E1:60_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/><div class="add"><a NAME="E2:60_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:60_1_1"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T12" onmouseover="rs('int_2/T12')" onmouseout="rh()">INT_2:12</a>, <a class="ref" href="rvsum_1.html#T94" onmouseover="rs('rvsum_1/T94')" onmouseout="rh()">RVSUM_1:94</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2"><span class="kw">suppose </span></a><a NAME="E1:60_1_2"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">f</font> being   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <font color="Olive" title="b1">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a>  holds <br/> <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Olive" title="b1">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> ;<br/><a NAME="E2:60_1_2"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 ;<br/><a NAME="E3:60_1_2"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_1_2_1"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c6">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp; <font color="Maroon" title="c6">f</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a>  implies  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c6">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:60_1_2_1"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <font color="Maroon" title="c6">f</font>
 <span class="kw">and </span><br/><a NAME="E3:60_1_2_1"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:60_1_2_1"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E5:60_1_2_1"/><span class="lab"><font color="Green" title="E61">A11</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E6:60_1_2_1"/><span class="lab"><font color="Green" title="E62">A12</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c6">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/>

<a NAME="E7:60_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_2_1"><span class="lab"><font color="Green" title="E58">A8</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:60_1_2_1"/><span class="lab"><font color="Green" title="E63">A13</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T46" onmouseover="rs('finseq_1/T46')" onmouseout="rh()">FINSEQ_1:46</a></span>;<br/>
<a NAME="E10:60_1_2_1"/>
<font color="Maroon" title="c7">g</font> is  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="number08.html#T3" target="_self" onmouseover="rs('number08/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">g</font> as   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E12:60_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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="c9">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_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"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E13:60_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="c9">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a></span>;<br/>
<a NAME="E14:60_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A15</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c9">g</font></span>)</span>;<br/>
<a NAME="E16:60_1_2_1"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_2_1"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E17:60_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A16</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E18:60_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A17</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:60_1_2_1"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <font color="Maroon" title="c9">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:60_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c6">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/><div class="add"><a NAME="E2:60_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="filerec1.html#T3" onmouseover="rs('filerec1/T3')" onmouseout="rh()">FILEREC1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_2_1_1_1"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c6">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_2_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E14:60_1_2_1"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="ref" href="newton.html#T35" onmouseover="rs('newton/T35')" onmouseout="rh()">NEWTON:35</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:60_1_2_1_1_2"/><span class="lab"><font color="Green" title="E68">A18</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <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> <font color="Maroon" title="c9">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c6">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/><div class="add"><a NAME="E2:60_1_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A19</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E3:60_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A20</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c9">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:60_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_1_2_1"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="txt" href="#E2:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A19</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:60_1_2_1_1_2"/><span class="lab"><font color="Green" title="E72">A22</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D4" target="_self" onmouseover="rs('number08/D4')" onmouseout="rh()">NUMBER08:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">g</font> implies <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E73">A23</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E2:60_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E74">A24</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="finseq_1.html#T26" onmouseover="rs('finseq_1/T26')" onmouseout="rh()">FINSEQ_1:26</a></span>;<br/>
<a NAME="E3:60_1_2_1_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="txt" href="#E1:60_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E73">A23</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:60_1_2_1_1_2_1"/>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_2_1"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E1:60_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E73">A23</font></span></a>, <a class="txt" href="#E2:60_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E74">A24</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E6:60_1_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A25</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is  <a href="valued_0.html#V2" title="VALUED_0:attr.2">ext-real-valued</a> 
 ;<br/><a NAME="E7:60_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A26</font></span>: 
<font color="Maroon" title="c9">g</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_2_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="number08.html#T36" target="_self" onmouseover="rs('number08/T36')" onmouseout="rh()">Th36</a></span>;<br/><a NAME="E8:60_1_2_1_1_2"/>
<font color="Maroon" title="c9">g</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_2_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="txt" href="#E6:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A25</font></span></a>, <a class="ref" href="number08.html#T37" target="_self" onmouseover="rs('number08/T37')" onmouseout="rh()">Th37</a></span>;<br/><a NAME="E9:60_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A27</font></span>: 
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c9">g</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_2_1"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E3:60_1_2_1"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E13:60_1_2_1"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="txt" href="#E1:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A18</font></span></a>, <a class="txt" href="#E5:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A22</font></span></a>, <a class="txt" href="#E7:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A26</font></span></a></span>;<br/><a NAME="E10:60_1_2_1_1_2"/><span class="lab"><font color="Green" title="E76">A28</font></span>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A27</font></span></a>, <a class="ref" href="nat_3.html#T1" onmouseover="rs('nat_3/T1')" onmouseout="rh()">NAT_3:1</a></span>;<br/><a NAME="E11:60_1_2_1_1_2"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_2_1"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="txt" href="#E18:60_1_2_1"><span class="lab"><font color="Green" title="E67">A17</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E13:60_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A20</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E14:60_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c9">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c9">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><a NAME="E15:60_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c9">g</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_2_1"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E13:60_1_2_1"><span class="lab"><font color="Green" title="E64">A14</font></span></a>, <a class="txt" href="#E14:60_1_2_1"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E17:60_1_2_1"><span class="lab"><font color="Green" title="E66">A16</font></span></a>, <a class="txt" href="#E2:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A19</font></span></a>, <a class="txt" href="#E4:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A21</font></span></a>, <a class="ref" href="valued_0.html#D13" onmouseover="rs('valued_0/D13')" onmouseout="rh()">VALUED_0:def 13</a>, <a class="ref" href="int_1.html#T52" onmouseover="rs('int_1/T52')" onmouseout="rh()">INT_1:52</a></span>;<br/><a NAME="E16:60_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c12">x1</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T6" target="_self" onmouseover="rs('number08/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E17:60_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A29</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T1" onmouseover="rs('nat_3/T1')" onmouseout="rh()">NAT_3:1</a></span>;<br/><a NAME="E18:60_1_2_1_1_2"/><span class="lab"><font color="Green" title="E78">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" onmouseover="rs('newton/T15')" onmouseout="rh()">NEWTON:15</a></span>;<br/><a NAME="E19:60_1_2_1_1_2"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_2_1"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="rvsum_1.html#T96" onmouseover="rs('rvsum_1/T96')" onmouseout="rh()">RVSUM_1:96</a></span>;<br/><span class="kw">hence </span><a NAME="E20:60_1_2_1_1_2"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c6">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:60_1_2_1"><span class="lab"><font color="Green" title="E65">A15</font></span></a>, <a class="txt" href="#E10:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A28</font></span></a>, <a class="txt" href="#E17:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A29</font></span></a>, <a class="txt" href="#E18:60_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A30</font></span></a>, <a class="ref" href="int_2.html#T9" onmouseover="rs('int_2/T9')" onmouseout="rh()">INT_2:9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E4:60_1_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:60_1_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="#E3:60_1_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>);<br/></span><a NAME="E5:60_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A31</font></span>: 
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E1:60_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a></span>;<br/><a NAME="E6:60_1_2"/>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E7:60_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E8:60_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="number08.html#T6" target="_self" onmouseover="rs('number08/T6')" onmouseout="rh()">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E9:60_1_2"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_2"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="ref" href="int_2.html#T9" onmouseover="rs('int_2/T9')" onmouseout="rh()">INT_2:9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

</div>
