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<b>let </b><font color="Maroon">f</font> be  <a href="series_1.html#V1">summable</a> <a href="seq_1.html#NM1">Real_Sequence</a>, <font color="Maroon">g</font> be  <a href="series_1.html#V1">summable</a> <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>

<a NAME="E1:9"/><i><font color="Green">E37</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">g</font> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E2:9"/><i><font color="Green">E38</font></i>: 
(  <a href="series_1.html#K2">Sum</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="series_1.html#K2">Sum</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="series_1.html#D3">SERIES_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:9"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E4:9"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="series_1.html#T17">SERIES_1:17</a></i>;<br/>
<b>hence </b><a NAME="E5:9"/>
 <a href="series_1.html#K2">Sum</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="series_1.html#K2">Sum</a> <font color="Maroon">g</font>
 <b>by </b><i>, , <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>


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