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<b>let </b><font color="Maroon" title="c1">s2</font>, <font color="Maroon" title="c2">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E19">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">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">s1</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 ;<br/>

<b>given </b><font color="Maroon" title="c3">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> <b> such that </b><a NAME="E3:21"/><i><font color="Green" title="E21">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>   st <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c1">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>
 ;<br/>

<div><i><font color="Green" title="E22">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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="E1:21_1"/><i><font color="Green" title="E23">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/><a NAME="E2:21_1"/><i><font color="Green" title="E24">A6</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="series_1.html#E1:21"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><a NAME="E3:21_1"/>
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E4:21_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E1:21_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="series_1.html#E2:21_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="series_1.html#E3:21"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:21"/>
<font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E2:21"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="series_1.html#T15" target="_self" title="SERIES_1:th.15">Th15</a></i>;<br/>
<a NAME="E6:21"/><b>then </b>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E4:21"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="series_1.html#T20" target="_self" title="SERIES_1:th.20">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:21"/><i><font color="Green" title="E23">A7</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></i>;<br/>
<div><i><font color="Green" title="E24">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="E1:21_2"/>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E2:21_2"/><b>then </b>
<font color="Maroon" title="c1">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="series_1.html#E3:21"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/><a NAME="E3:21_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/><b>hence </b><a NAME="E4:21_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E25">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="E1:21_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/><b>hence </b><a NAME="E2:21_3"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E1:21"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="E1:21_4"/><i><font color="Green" title="E26">A10</font></i>: 
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="series_1.html#E8:21"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/><a NAME="E2:21_4"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="series_1.html#E9:21"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="series_1.html#T17" target="_self" title="SERIES_1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E3:21_4"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="series_1.html#E1:21_4"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:21"/><b>then </b>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font></span>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D3" title="SEQ_2:def.3">SEQ_2:def 3</a></i>;<br/>
<a NAME="E13:21"/><b>then </b>
<font color="Maroon" title="c1">s2</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c3">m</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E10:21"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="series_1.html#T20" target="_self" title="SERIES_1:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E14:21"/>
<font color="Maroon" title="c1">s2</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="series_1.html#T16" target="_self" title="SERIES_1:th.16">Th16</a></i>;<br/>


</div>
