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

<b>let </b><font color="Maroon" title="c1">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">n</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   <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="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>

<a NAME="E2:4"/><i><font color="Green" title="E5">A2</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="series_1.html#T19" title="SERIES_1:th.19">SERIES_1:19</a></i>;<br/>
<b>thus </b><a NAME="E3:4"/><i><font color="Green" title="E6">A3</font></i>: 
for <font color="Olive" title="b1">n</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   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</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/>

<a NAME="E1:4_1"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E2:4_1"/><i><font color="Green" title="E8">A5</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>;<br/>
<a NAME="E3:4_1"/>
 <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="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="rsspace2.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:4_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="rsspace2.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="rsspace2.html#E1:4_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="rsspace2.html#E2:4_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="seqm_3.html#T11" title="SEQM_3:th.11">SEQM_3:11</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:4"/><i><font color="Green" title="E7">A6</font></i>: 
for <font color="Olive" title="b1">n</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" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_1"><b>case </b></a><a NAME="E1:4_2_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_2_1_1_1"/>
<font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1_2"><b>case </b></a><a NAME="E1:4_2_1_1_2"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:4_2_1_1_2"/>
 <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="c2">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E3:4_2_1_1_2"/><b>then </b><i><font color="Green" title="E9">A8</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="rsspace2.html#E1:4_2_1_1_2"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>set </b><font color="Maroon" title="c3">nn</font> = <font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1;<br/><a NAME="E4:4_2_1_1_2"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="rsspace2.html#E3:4_2_1_1_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E5:4_2_1_1_2"/><i><font color="Green" title="E11">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E6:4_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rsspace2.html#E3:4"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="rsspace2.html#E4:4_2_1_1_2"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/><a NAME="E7:4_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E8:4_2_1_1_2"/>
<font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="rsspace2.html#E4:4_2_1_1_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="rsspace2.html#E5:4_2_1_1_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="series_1.html#D1" title="SERIES_1:def.1">SERIES_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_2"/>
<font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:4"/>
<font color="Maroon" title="c1">rseq</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ;<br/>

<a NAME="E6:4"/><b>then </b><i><font color="Green" title="E8">A11</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="series_1.html#T20" title="SERIES_1:th.20">SERIES_1:20</a></i>;<br/>
<b>thus </b><a NAME="E7:4"/><i><font color="Green" title="E9">A12</font></i>: 
for <font color="Olive" title="b1">n</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  <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">rseq</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</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/>

<a NAME="E1:4_3"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="rsspace2.html#E6:4"><i><font color="Green" title="E8">A11</font></i></a>, <a class="ref" href="seq_4.html#T54" title="SEQ_4:th.54">SEQ_4:54</a></i>;<br/>
<b>hence </b><a NAME="E2:4_3"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">rseq</font>
 <b>by </b><i><a class="ref" href="series_1.html#D3" title="SERIES_1:def.3">SERIES_1:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c2">n</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/>

<a NAME="E8:4"/><i><font color="Green" title="E10">A13</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">rseq</font>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E7:4"><i><font color="Green" title="E9">A12</font></i></a></i>;<br/>
<a NAME="E9:4"/>
<font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">rseq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="rsspace2.html#E4:4"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:4"/>
<font color="Maroon" title="c1">rseq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">rseq</font>
 <b>by </b><i><a class="txt" href="rsspace2.html#E8:4"><i><font color="Green" title="E10">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div>
