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

<b>let </b><font color="Maroon" title="c3">th</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:78"/><i><font color="Green" title="E64">A1</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">th</font> &amp; <font color="Maroon" title="c3">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> )
 ;<br/>

<a NAME="E2:78"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a>  is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<a NAME="E1:78_1"/>
for <font color="Olive" title="b1">m</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>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> $1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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>  );<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">th</font>
<b>by </b><i><a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></i>
<br/>.= 
1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">th</font>
<b>by </b><i><a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></i>
<br/>.= 
<font color="Maroon" title="c3">th</font>

;<br/>
<a NAME="E2:78_1_1"/><b>then </b><i><font color="Green" title="E66">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:78"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></i>;<br/>
<a NAME="E3:78_1_1"/><i><font color="Green" title="E67">A4</font></i>: 
for <font color="Olive" title="b1">m</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>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:78_1_1_2"/><i><font color="Green" title="E68">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<a NAME="E2:78_1_1_2"/><i><font color="Green" title="E69">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">th</font>
 
<b>by </b><i><a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></i>;<br/>
<a NAME="E3:78_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">th</font>
 
<b>by </b><i><a class="ref" href="prepower.html#T4" title="PREPOWER:th.4">PREPOWER:4</a></i>;<br/>
<a NAME="E4:78_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="sin_cos.html#E1:78"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sin_cos.html#E1:78_1_1_2"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/>
<b>hence </b><a NAME="E5:78_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:78"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sin_cos.html#E2:78_1_1_2"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="real_2.html#T147" title="REAL_2:th.147">REAL_2:147</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:78_1_1"/>
for <font color="Olive" title="b1">m</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="sin_cos.html#E2:78_1_1"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="sin_cos.html#E3:78_1_1"><i><font color="Green" title="E67">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:78_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:78_1"/>
<font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a>  is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D14" title="SEQM_3:def.14">SEQM_3:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:78"/><i><font color="Green" title="E66">A7</font></i>: 
<font color="Maroon" title="c3">th</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>;<br/>
<a NAME="E4:78"/>
<font color="Maroon" title="c3">th</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">th</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:78"/>
<font color="Maroon" title="c3">th</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">th</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:78"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="sin_cos.html#E2:78"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="sin_cos.html#E3:78"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="seqm_3.html#T14" title="SEQM_3:th.14">SEQM_3:14</a></i>;<br/>


</div>
