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

<b>let </b><font color="Maroon">Fn</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:68"/><i><font color="Green">E38</font></i>: 
( (  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 ) implies  <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E39</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_1"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1_1"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_1"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4" target="_self">"**"</a> <font color="Maroon">Fn</font> )
 ;<br/><b>hence </b><a NAME="E3:68_1_1_1_1"/>
 <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="binop_2.html#T5">BINOP_2:5</a>, <a class="ref" href="stirl2_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_1_2"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_2"/><b>then </b>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E3:68_1_1_1_2"/><b>then </b>
(  <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font></span>)</span> <a href="binop_2.html#K24">*</a> 0 &amp;  <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font></span>)</span> <a href="binop_2.html#K24">*</a> 0 )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:68_1_1_1_2"/>
 <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_1"/>
 <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:68"/>
(  not  <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_2"/><i><font color="Green">E40</font></i>: 
 <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_2_1_1"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_2_1_1_1"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_2_1_1_1"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_2_1_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>assume </b><a NAME="E2:68_2_1_1_2"/>
( not  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 &amp;  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> &amp; not <font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:68_2_1_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> &amp; <font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E5:68_2_1_1_2"/>
 <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T12" target="_self">Th12</a>, <a class="txt" href="stirl2_1.html#E1:68_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="stirl2_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E6:68_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="stirl2_1.html#T10" target="_self">Th10</a>, <a class="txt" href="stirl2_1.html#E1:68_2_1_1_2"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_2"/>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68"/>
(  <a href="stirl2_1.html#K7" target="_self">Sum</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 iff (  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">Fn</font> <a href="hidden.html#R1">=</a> 0 or for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">Fn</font> holds <br/><font color="Maroon">Fn</font> <a href="stirl2_1.html#K8" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/>


</div>
