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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  implies ( <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>  &amp;  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:13"/>
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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>  &amp;  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )</span><br/>

<a NAME="E2:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A1</font></span>: 
(  <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>  &amp;  <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D2" target="_self" title="SERIES_1:def.2">Def2</a></span>;<br/>
<a NAME="E3:13"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></span>;<br/>
<a NAME="E4:13"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T9" target="_self" title="SERIES_1:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:13"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#D2" target="_self" title="SERIES_1:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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></span><br/>

<span class="kw">thus </span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T9" target="_self" title="SERIES_1:th.9">Th9</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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