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<b>let </b><font color="Maroon" title="c3">seq</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"> for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> holds <br/>( <font color="Olive" title="b1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp;  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c4">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> implies ( <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp;  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span> ) )</span><br/>



<b>assume </b><a NAME="E1:81"/>
<font color="Maroon" title="c3">seq</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="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp;  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span> )</span><br/>

<a NAME="E2:81"/><b>then </b><i><font color="Green" title="E49">A1</font></i>: 
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10" target="_self" title="COMSEQ_3:def.10">Def10</a></i>;<br/>
<a NAME="E3:81"/><b>then </b><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_2.html#T17" title="COMSEQ_2:th.17">COMSEQ_2:17</a></i>;<br/>
<a NAME="E4:81"/><i><font color="Green" title="E51">A3</font></i>: 
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_3.html#T29" target="_self" title="COMSEQ_3:th.29">Th29</a></i>;<br/>
 <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font></span>)</span> = 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#T29" target="_self" title="COMSEQ_3:th.29">Th29</a></i>
<br/>.= 
<font color="Maroon" title="c4">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E2:81"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="comseq_2.html#T18" title="COMSEQ_2:th.18">COMSEQ_2:18</a></i>
;<br/>
<b>hence </b><a NAME="E6:81"/>
( <font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> &amp;  <a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" title="COMSEQ_3:func.8">Sum</a> <font color="Maroon" title="c3">seq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="comseq_3.html#E3:81"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="comseq_3.html#E4:81"><i><font color="Green" title="E51">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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