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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="clopban2.html#NM3">Complex_Banach_Algebra</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="E1:4"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_1.html#V6">convergent</a>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="clvect_1.html#V6">convergent</a>
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E7</font></i>: 
 <a href="clvect_1.html#K11">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="clvect_1.html#K7">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:4"/>
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</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="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="clopban4.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="clopban4.html#E2:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="clopban4.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="clvect_1.html#T122">CLVECT_1:122</a></i>;<br/>
<a NAME="E5:4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>;<br/>
<a NAME="E6:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="clopban3.html#T42">CLOPBAN3:42</a></i>;<br/>
<a NAME="E7:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="clvect_1.html#K11">lim</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="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/>
<b>hence </b><a NAME="E8:4"/>
 <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="clvect_1.html#K11">lim</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/>


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