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



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c1">s1</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> &amp; <font color="Maroon" title="c2">s2</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> )
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c1">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2" target="_self" title="SERIES_1:def.2">Def2</a></i>;<br/>
<a NAME="E3:10"/><i><font color="Green" title="E12">A3</font></i>: 
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s2</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="series_1.html#D2" target="_self" title="SERIES_1:def.2">Def2</a></i>;<br/>
<a NAME="E4:10"/><b>then </b>
<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">s1</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</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="c2">s2</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="series_1.html#E2:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/>
<a NAME="E5:10"/><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">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c2">s2</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T9" target="_self" title="SERIES_1:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E6:10"/>
<font color="Maroon" title="c1">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c2">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#D2" target="_self" title="SERIES_1:def.2">Def2</a></i>;<br/>

<b>thus </b> <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c2">s2</font></span>)</span> = 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</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">s1</font></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <font color="Maroon" title="c2">s2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#T9" target="_self" title="SERIES_1:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c1">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <font color="Maroon" title="c2">s2</font></span>)</span>
<b>by </b><i><a class="txt" href="series_1.html#E2:10"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="series_1.html#E3:10"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>
;<br/>


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