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<b>let </b><font color="Maroon" title="c3">seq1</font>, <font color="Maroon" title="c4">seq2</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">seq1</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> implies ( <font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp;  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <font color="Maroon" title="c4">seq2</font></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:74"/>
( <font color="Maroon" title="c3">seq1</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp; <font color="Maroon" title="c4">seq2</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp;  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <font color="Maroon" title="c4">seq2</font></span>)</span> )</span><br/>

<a NAME="E2:74"/><b>then </b><i><font color="Green" title="E44">A1</font></i>: 
(  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c3">seq1</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c4">seq2</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> )
 
;<br/>
<a NAME="E3:74"/><b>then </b><i><font color="Green" title="E45">A2</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c4">seq2</font></span>)</span> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#T13" title="COMSEQ_2:th.13">COMSEQ_2:13</a></i>;<br/>
<a NAME="E4:74"/><i><font color="Green" title="E46">A3</font></i>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T27" target="_self" title="COMSEQ_3:th.27">Th27</a></i>;<br/>
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font></span>)</span> = 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T27" target="_self" title="COMSEQ_3:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c3">seq1</font></span>)</span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c4">seq2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:74"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="comseq_2.html#T14" title="COMSEQ_2:th.14">COMSEQ_2:14</a></i>
;<br/>
<b>hence </b><a NAME="E6:74"/>
( <font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp;  <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq1</font> <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <font color="Maroon" title="c4">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <font color="Maroon" title="c3">seq1</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <font color="Maroon" title="c4">seq2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:74"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E4:74"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="comseq_3.html#D10" target="_self" title="COMSEQ_3:def.10">Def10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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