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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_3.html#V2" target="_self">absolutely_summable</a>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="clvect_3.html#V2" target="_self">absolutely_summable</a>
 ;<br/>

<a NAME="E3:30"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="clvect_3.html#E1:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="clvect_3.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:30"/>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_2.html#K2">.||</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="clvect_3.html#E2:30"><i><font color="Green">E18</font></i></a>, <a class="ref" href="clvect_3.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E5:30"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_2.html#K2">.||</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="clvect_3.html#E3:30"><i><font color="Green">E19</font></i></a>, <a class="ref" href="series_1.html#T10">SERIES_1:10</a></i>;<br/>
<a NAME="E6:30"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><a NAME="E1:30_1"/>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1"/>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="csspace.html#K13">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1"/>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="csspace.html#T46">CSSPACE:46</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:30_1"/>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D11">CLVECT_1:def 11</a></i>
;<br/>
<a NAME="E2:30_1_2"/><b>then </b>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="csspace.html#K13">.||</a></span>
 
<b>by </b><i><a class="ref" href="csspace.html#T48">CSSPACE:48</a></i>;<br/>
<a NAME="E3:30_1_2"/><b>then </b>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="csspace.html#K13">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>;<br/>
<a NAME="E4:30_1_2"/><b>then </b>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="clvect_2.html#D3">CLVECT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_2"/>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_2.html#K2">.||</a></span> <a href="seq_1.html#K9">+</a> <span class="p2"><a href="clvect_2.html#K2">||.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="clvect_2.html#K2">.||</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="seq_1.html#T11">SEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E7:30"/><b>then </b>
<span class="p1"><a href="clvect_2.html#K2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="clvect_2.html#K2">.||</a></span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="txt" href="clvect_3.html#E5:30"><i><font color="Green">E20</font></i></a>, <a class="ref" href="series_1.html#T24">SERIES_1:24</a></i>;<br/>
<b>hence </b><a NAME="E8:30"/>
<font color="Maroon">c<sub>2</sub></font> <a href="csspace.html#K18">+</a> <font color="Maroon">c<sub>3</sub></font> is <a href="clvect_3.html#V2" target="_self">absolutely_summable</a>
 <b>by </b><i><a class="ref" href="clvect_3.html#D8" target="_self">Def8</a></i>;<br/>


</div>
