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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be  <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">z</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<a NAME="E1:41"/><i><font color="Green" title="E33">A1</font></i>: 
 <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="lopban_3.html#D2" target="_self" title="LOPBAN_3:def.2">Def2</a></i>;<br/>
<a NAME="E2:41"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T37" title="NORMSP_1:th.37">NORMSP_1:37</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green" title="E35">A3</font></i>: 
 <a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">z</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lopban_3.html#T24" target="_self" title="LOPBAN_3:th.24">Th24</a></i>;<br/>
 <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> = 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <span class="p2">(<span class="default"><a href="lopban_3.html#K1" title="LOPBAN_3:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#T24" target="_self" title="LOPBAN_3:th.24">Th24</a></i>
<br/>.= 
<font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2" title="LOPBAN_3:func.2">Sum</a> <font color="Maroon" title="c2">seq</font></span>)</span>
<b>by </b><i><a class="txt" href="lopban_3.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="normsp_1.html#T45" title="NORMSP_1:th.45">NORMSP_1:45</a></i>
;<br/>
<b>hence </b><a NAME="E5:41"/>
( <font color="Maroon" title="c3">z</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c2">seq</font> is <a href="lopban_3.html#V1" title="LOPBAN_3:attr.1">summable</a> &amp;  <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="normsp_1.html#K6" title="NORMSP_1:func.6">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="lopban_3.html#K2" title="LOPBAN_3:func.2">Sum</a> <font color="Maroon" title="c2">seq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="lopban_3.html#E2:41"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="lopban_3.html#E3:41"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="lopban_3.html#D2" target="_self" title="LOPBAN_3:def.2">Def2</a></i>;<br/>


</div>
