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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"> ( <font color="Maroon" title="c3">seq</font> is <a href="comseq_3.html#V1" title="COMSEQ_3:attr.1">summable</a> implies for <font color="Olive" title="b1">z</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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#K7" title="COMSEQ_3:func.7">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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">seq</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:77"/>
<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"> for <font color="Olive" title="b1">z</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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#K7" title="COMSEQ_3:func.7">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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">seq</font></span>)</span> )</span><br/>

<a NAME="E2:77"/><b>then </b><i><font color="Green" title="E46">A1</font></i>: 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c3">seq</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>
 
;<br/>
<b>let </b><font color="Maroon" title="c4">z</font> be  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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#K7" title="COMSEQ_3:func.7">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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">seq</font></span>)</span> )</span><br/>

<a NAME="E3:77"/><i><font color="Green" title="E47">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#K6" title="COMSEQ_3:func.6">Partial_Sums</a> <font color="Maroon" title="c3">seq</font></span>)</span> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>
 
<b>by </b><i><a class="txt" href="#E2:77"><i><font color="Green" title="E46">A1</font></i></a></i>;<br/>
<a NAME="E4:77"/><i><font color="Green" title="E48">A3</font></i>: 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">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#K6" title="COMSEQ_3:func.6">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#K7" title="COMSEQ_3:func.7">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#K6" title="COMSEQ_3:func.6">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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">seq</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:77"><i><font color="Green" title="E46">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="E6:77"/>
( <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#K7" title="COMSEQ_3:func.7">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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">seq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E3:77"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="#E4:77"><i><font color="Green" title="E48">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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