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<div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:94"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 ;<br/>

<a NAME="E2:94"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K6">1r</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#T134">COMPLEX1:134</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K2" target="_self">#N</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E1:94"><i><font color="Green">E56</font></i></a>, <a class="txt" href="comseq_3.html#E2:94"><i><font color="Green">E57</font></i></a>, <a class="ref" href="comseq_3.html#T37" target="_self">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E2:94"><i><font color="Green">E57</font></i></a>, <a class="ref" href="comseq_3.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="comseq_1.html#D7">COMSEQ_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:94"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="comseq_1.html#K4">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="comseq_1.html#T6">COMSEQ_1:6</a></i>;<br/>
<a NAME="E5:94"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a>  is <a href="comseq_3.html#V1" target="_self">summable</a> &amp;  <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="hidden.html#R1">=</a> <a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="comseq_3.html#E1:94"><i><font color="Green">E56</font></i></a>, <a class="ref" href="comseq_3.html#T65" target="_self">Th65</a></i>;<br/>
<a NAME="E6:94"/><b>then </b><i><font color="Green">E59</font></i>: 
(  <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E7:94"/><b>then </b>
(  <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="comseq_3.html#E4:94"><i><font color="Green">E58</font></i></a>, <a class="ref" href="comseq_2.html#T17">COMSEQ_2:17</a>, <a class="ref" href="comseq_2.html#T18">COMSEQ_2:18</a></i>;<br/>
<b>hence </b><a NAME="E8:94"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="comseq_3.html#V1" target="_self">summable</a>
 <b>by </b><i><a class="ref" href="comseq_3.html#D10" target="_self">Def10</a></i>;<br/>

 <a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="comseq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="comseq_3.html#K1" target="_self">GeoSeq</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E4:94"><i><font color="Green">E58</font></i></a>, <a class="txt" href="comseq_3.html#E6:94"><i><font color="Green">E59</font></i></a>, <a class="ref" href="comseq_2.html#T18">COMSEQ_2:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="comseq_3.html#E1:94"><i><font color="Green">E56</font></i></a>, <a class="ref" href="comseq_3.html#T65" target="_self">Th65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K9">*</a> <a href="complex1.html#K6">1r</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E10:94"/>
 <a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="comseq_1.html#K1">.</a> 0</span>)</span> <a href="complex1.html#K13">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K11">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


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