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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</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="E1:10"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="bhsp_4.html#V1" target="_self">summable</a>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_4.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E3:10"/><i><font color="Green">E8</font></i>: 
 <a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font> is <a href="bhsp_2.html#V1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E1:10"><i><font color="Green">E6</font></i></a>, <a class="ref" href="bhsp_4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E9</font></i>: 
 <a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_2.html#V1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E2:10"><i><font color="Green">E7</font></i></a>, <a class="ref" href="bhsp_4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:10"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="bhsp_2.html#V1">convergent</a>
 
<b>by </b><i><a class="txt" href="bhsp_4.html#E3:10"><i><font color="Green">E8</font></i></a>, <a class="ref" href="bhsp_2.html#T4">BHSP_2:4</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
 <a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="bhsp_2.html#V1">convergent</a>
 
<b>by </b><i><a class="ref" href="bhsp_4.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:10"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="bhsp_4.html#V1" target="_self">summable</a>
 <b>by </b><i><a class="ref" href="bhsp_4.html#D2" target="_self">Def2</a></i>;<br/>

<b>thus </b> <a href="bhsp_4.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="bhsp_2.html#K1">lim</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="bhsp_2.html#K1">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="normsp_1.html#K4">-</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bhsp_4.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_4.html#E3:10"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bhsp_4.html#E4:10"><i><font color="Green">E9</font></i></a>, <a class="ref" href="bhsp_2.html#T14">BHSP_2:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K1">lim</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="bhsp_4.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K2" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>


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