<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">seq1</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp; <font color="Olive" title="b2">seq2</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  holds <br/>( <font color="Olive" title="b1">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Olive" title="b2">seq2</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp;  <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Olive" title="b2">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Olive" title="b1">seq1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Olive" title="b2">seq2</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">seq1</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp; <font color="Maroon" title="c3">seq2</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  implies ( <font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp;  <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:23"/>
( <font color="Maroon" title="c2">seq1</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp; <font color="Maroon" title="c3">seq2</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp;  <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span> )</span><br/>

<a NAME="E2:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A1</font></span>: 
(  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  &amp;  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a>  )
 
;<br/>
<a NAME="E3:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span> is  <a href="clvect_1.html#V9" title="CLVECT_1:attr.9">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_1.html#T113" onmouseover="rs('clvect_1/T113')" onmouseout="rh()">CLVECT_1:113</a></span>;<br/>
<a NAME="E4:23"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T15" target="_self" onmouseover="rs('clopban3/T15')" onmouseout="rh()">Th15</a></span>;<br/>
 <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> = 
 <a href="clvect_1.html#K7" title="CLVECT_1:func.7">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T15" target="_self" onmouseover="rs('clopban3/T15')" onmouseout="rh()">Th15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="clvect_1.html#K7" title="CLVECT_1:func.7">lim</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K7" title="CLVECT_1:func.7">lim</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="clvect_1.html#T119" onmouseover="rs('clvect_1/T119')" onmouseout="rh()">CLVECT_1:119</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:23"/>
( <font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font> is  <a href="clopban3.html#V1" title="CLOPBAN3:attr.1">summable</a>  &amp;  <a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="normsp_1.html#K1" title="NORMSP_1:func.1">+</a> <font color="Maroon" title="c3">seq2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="clopban3.html#K1" title="CLOPBAN3:func.1">Sum</a> <font color="Maroon" title="c3">seq2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
