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

<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E7">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  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>

<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/>

<b>assume </b><a NAME="E2:9"/><i><font color="Green" title="E8">A2</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">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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="E1:9_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E2:9_1"/>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">a</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="series_1.html#T19" title="SERIES_1:th.19">SERIES_1:19</a></i>;<br/><a NAME="E3:9_1"/><b>then </b><i><font color="Green" title="E10">A4</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">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</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">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:9_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="seqm_3.html#T12" title="SEQM_3:th.12">SEQM_3:12</a></i>;<br/><a NAME="E4:9_1"/>
<font color="Maroon" title="c1">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</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">a</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="holder_1.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="holder_1.html#E6"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon" title="c1">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="holder_1.html#E1:9"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="holder_1.html#E2:9"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="holder_1.html#E3:9_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>


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